A Subclass of Analytic Functions Related to -Uniformly Convex...
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Research Article A Subclass of Analytic Functions Related to -Uniformly Convex and Starlike Functions Saqib Hussain, 1 Akhter Rasheed, 2 Muhammad Asad Zaighum, 2 and Maslina Darus 3 1 Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan 2 Department of Mathematics & Statistics, Riphah International University, Islamabad, Pakistan 3 School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia Correspondence should be addressed to Saqib Hussain; [email protected] Received 26 January 2017; Accepted 20 April 2017; Published 23 May 2017 Academic Editor: Maria Alessandra Ragusa Copyright © 2017 Saqib Hussain et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate some subclasses of -uniformly convex and -uniformly starlike functions in open unit disc, which is generalization of class of convex and starlike functions. Some coefficient inequalities, a distortion theorem, the radii of close-to-convexity, and starlikeness and convexity for these classes of functions are studied. e behavior of these classes under a certain modified convolution operator is also discussed. 1. Introduction Let A be the class of all analytic functions in open unit disc Δ = { : || < 1}, normalized by (0) = 0 and (0) = 1. us, any ∈ A has the following Maclaurin’s series: () = + ∞ ∑ =2 . (1) A function is said to be univalent if it never takes same value twice. By S we mean the subclass of A which is com- posed of univalent functions. By ST and CV we mean the well-known subclasses of A that are, respectively, starlike and convex. In 1991, Goodman [1, 2] introduced the classes UCV and UST of uniformly convex and uniformly starlike functions, respectively. A function ∈ is uniformly convex if () maps every circular arc contained in Δ with center ∈Δ onto a convex arc. e function ∈ is uniformly starlike if () maps every circular arc contained in Δ with center ∈Δ onto a starlike arc with respect to (). A more useful representation of UCV and UST was given in [3–6] as ∈ UCV ⇐⇒∈ A, Re ( ( ()) () )> () () , ∈ Δ. ∈ UST ⇐⇒∈ A, Re ( () () )> () () −1 , ∈ Δ. (2) In 1999, for ≥0, Kanas and Wisniowska [7] introduced the class − UCV and − UST as ∈− UCV ⇐⇒ ∈− UST ⇐⇒∈ A, Re ( ( ()) () )> () () , ∈ Δ. (3) Observe that 0− UCV ≡ CV, 0− UST ≡ UST and 1− UCV ≡ UCV, 1− UST ≡ UST. For fixed ≥0, these classes have a nice geometrical representation; for detail see [7–9]. Hindawi Journal of Function Spaces Volume 2017, Article ID 9010964, 7 pages https://doi.org/10.1155/2017/9010964
Transcript of A Subclass of Analytic Functions Related to -Uniformly Convex...
Research ArticleA Subclass of Analytic Functions Related to 119896-UniformlyConvex and Starlike Functions
Saqib Hussain1 Akhter Rasheed2 Muhammad Asad Zaighum2 andMaslina Darus3
1Department of Mathematics COMSATS Institute of Information Technology Abbottabad Pakistan2Department of Mathematics amp Statistics Riphah International University Islamabad Pakistan3School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia43600 Bangi Selangor Malaysia
Correspondence should be addressed to Saqib Hussain saqib_mathyahoocom
Received 26 January 2017 Accepted 20 April 2017 Published 23 May 2017
Academic Editor Maria Alessandra Ragusa
Copyright copy 2017 Saqib Hussain et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We investigate some subclasses of 119896-uniformly convex and 119896-uniformly starlike functions in open unit disc which is generalizationof class of convex and starlike functions Some coefficient inequalities a distortion theorem the radii of close-to-convexity andstarlikeness and convexity for these classes of functions are studied The behavior of these classes under a certain modifiedconvolution operator is also discussed
1 Introduction
LetA be the class of all analytic functions 119891 in open unit discΔ = 119911 |119911| lt 1 normalized by 119891(0) = 0 and 1198911015840(0) = 1Thus any 119891 isin A has the following Maclaurinrsquos series
119891 (119911) = 119911 + infinsum119899=2
119886119899119911119899 (1)
A function 119891 is said to be univalent if it never takes samevalue twice By S we mean the subclass of A which is com-posed of univalent functions By ST and CV we mean thewell-known subclasses ofA that are respectively starlike andconvex
In 1991 Goodman [1 2] introduced the classesUCV andUST of uniformly convex and uniformly starlike functionsrespectively A function 119891 isin 119878 is uniformly convex if 119891(119911)maps every circular arc 120574 contained in Δ with center 120577 isin Δonto a convex arc The function 119891 isin 119878 is uniformly starlikeif 119891(119911) maps every circular arc 120574 contained in Δ with center120577 isin Δ onto a starlike arc with respect to 119891(120577) A more usefulrepresentation ofUCV andUST was given in [3ndash6] as119891 isin UCVlArrrArr 119891 isin A
Re((1199111198911015840 (119911))10158401198911015840 (119911) ) gt 100381610038161003816100381610038161003816100381610038161003816 11991111989110158401015840 (119911)1198911015840 (119911)
100381610038161003816100381610038161003816100381610038161003816
119911 isin Δ119891 isin USTlArrrArr 119891 isin A
Re(1199111198911015840 (119911)119891 (119911) ) gt100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1
100381610038161003816100381610038161003816100381610038161003816 119911 isin Δ
(2)
In 1999 for 119896 ge 0 Kanas and Wisniowska [7] introducedthe class 119896 minusUCV and 119896 minusUST as
119891 isin 119896 minusUCVlArrrArr1199111198911015840 isin 119896 minusUSTlArrrArr 119891 isin A
Re((1199111198911015840 (119911))10158401198911015840 (119911) ) gt 119896 100381610038161003816100381610038161003816100381610038161003816 11991111989110158401015840 (119911)1198911015840 (119911)
100381610038161003816100381610038161003816100381610038161003816 119911 isin Δ
(3)
Observe that 0 minusUCV equiv CV 0 minusUST equiv UST and1 minusUCV equiv UCV 1 minusUST equiv USTFor fixed 119896 ge 0 these classes have a nice geometrical
representation for detail see [7ndash9]
HindawiJournal of Function SpacesVolume 2017 Article ID 9010964 7 pageshttpsdoiorg10115520179010964
2 Journal of Function Spaces
A lot of authors obtain very useful properties of UCVand UST and their generalization in several direction forexample see [1 2 7 8 10 11] and reference cited therein
For (0 le 120572 lt 1) in [4] (see also [12]) Ronning introducedthe following two important subclasses 119896 minusUST(120572) and 119896 minusUCV(120572) as
119891 isin 119896 minusUST (120572) lArrrArrRe1199111198911015840 (119911)119891 (119911) minus 120572 gt 119896
100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1
100381610038161003816100381610038161003816100381610038161003816 119911 isin Δ
119891 isin 119896 minusUCV (120572) lArrrArr1199111198911015840 isin 119896 minusUST (120572)
(4)
Recently in [13] El-Ashwah et al introduced two impor-tant subclass 119896 minus UCV(120572 120573) and 119896 minus UST(120572 120573) of 119896-uniformly convex starlike functions as
119891 isin 119896 minusUCV (120572 120573) lArrrArrRe(1199111198911015840 (119911))10158401198911015840 (119911) minus 120572 gt 119896
100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1199111198911015840 (119911))10158401198911015840 (119911) minus 120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119911 isin Δ119891 isin 119896 minusUST (120572 120573) lArrrArr
Re1199111198911015840 (119911)119891 (119911) minus 120572 gt 119896100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 120573
100381610038161003816100381610038161003816100381610038161003816 119911 isin Δ
(5)
where (0 le 120572 lt 120573 le 1) and 119896(1 minus 120573) lt 1 minus 120572Let 119891119895 (119895 = 1 2 ) be defined by
119891119895 (119911) = 119911 + infinsum119899=2
119886119899119895119911119899 119886119899119895 ge 0 119895 isin 119873 (6)
then the modified Hadmard product of 1198911(119911) and 1198912(119911) isdefined by
(1198911 lowast 1198912) (119911) = 119911 minus infinsum119899=2
11988611989911198861198992119911119899 (7)
We denote T by subclass of S consisting of functionshaving all negative coefficients in their Maclaurinrsquos seriesexpansions so any 119891 isin T has a series of the form
119891 (119911) = 119911 minus infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816 119911119899 119911 isin Δ (8)
Let V120578 be the class of functions 119891 isin S given in (1) forwhich arg(119886119899) = 120587 + (119899 minus 1)120578 119899 ge 2 Note thatV0 = T [11]
In recent years more and more researchers are interestedin the above defined classes (see [9 11 14ndash22])
In this paper by taking inspiration from the abovecited paper we introduce some new subclasses of analyticfunctions and obtain some interesting results
Definition 1 For (0 le 120572 lt 120573 le 1) 0 le 120575 lt 1 119896(1minus120573) lt 1minus120572and 0 le 120582 lt 1 a function 119891 isin S is in class 119896 minusU(120572 120573 120582 120575) ifand only if
Re(1 minus 120575) 1199111198911015840 + 120575 (1199111198911015840 + (1 + 2120582) 119911211989110158401015840 + 1205821199113119891101584010158401015840)(1 minus 120575) 119891 + 120575 (1199111198911015840 + 120582119911211989110158401015840)minus 120572ge 119896 10038161003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 120575) 1199111198911015840 + 120575 (1199111198911015840 + (1 + 2120582) 119911211989110158401015840 + 1205821199113119891101584010158401015840)(1 minus 120575) 119891 + 120575 (1199111198911015840 + 120582119911211989110158401015840)minus 12057310038161003816100381610038161003816100381610038161003816100381610038161003816 119911 isin Δ
(9)
Also
119896 minusVU120578 (120572 120573 120582 120575) = 119896 minusU (120572 120573 120582 120575) capV120578 (10)
It is worth mentioning that for special values of parame-ters these classes were extensively studied by many authorshere we mention few of them
(1) 119896 minusU(120572 120573 120582 1) = 119896 minus 119880(120582 120573 120572) [21](2) 119896 minusVU0(120572 120573 120582 1) = 119896 minus 119881119880120578(120582 120573 120572) [21](3) 0 minusVU0(120572 1 0 1) = CV(120572) [11](4) 119896 minusVU0(120572 1 0 1) = 119896 minusUCV(120572) [23](5) 1 minusU(120572 1 0 1) = UCV(120572) [4](6) 119896 minusU(120572 120573 0 0) = 119896 minusUST(120572 120573) [13]Throughout the paper minus1 le 120572 le 120573 le 1 0 le 120582 lt 11 minus 120572 gt 119896(1 minus 120573) and 119911 isin Δ unless otherwise stated
2 Main Results
Theorem 2 A function 119891(119911) given by (1) is in class 119896 minusU(120572 120573 120582 120575) if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(11)
where
Π119899 = (119899 + 120575119899 (119899 minus 1) (1 + 120582119899)) Ω119899 = 1 minus 120575 + 120575119899 (1 + 120582 (119899 minus 1)) (12)
Proof It is sufficient to prove that inequality (9) holds As weknow
Re (119908) gt 119896 1003816100381610038161003816119908 minus 1205731003816100381610038161003816 + 120572iff Re ((1 + 119896119890119894120579)119908 minus 120573119896119890119894120579) ge 120572 (13)
Journal of Function Spaces 3
then inequality (9) can be written as
Re((1 + 119896119890119894120579)sdot (1 minus 120575) 1199111198911015840 + 120575 (1199111198911015840 + (1 + 2120582) 119911211989110158401015840 + 1205821199113119891101584010158401015840)(1 minus 120575) 119891 + 120575 (1199111198911015840 + 120582119911211989110158401015840)minus 120573119896119890119894120579) ge 120572
(14)
This is
Re(119860 (119911)119861 (119911) ) ge 120572 (15)
where
119860 (119911)= (1 + 119896119890119894120579) (1199111198911015840 + 120575 ((1 + 2120582) 119911211989110158401015840 + 1205821199113119891101584010158401015840))minus 120573119896119890119894120579 ((1 minus 120575) 119891 + 120575 (1199111198911015840 + 120582119911211989110158401015840))
119861 (119911) = (1 minus 120575) 119891 + 120575 (1199111198911015840 + 120582119911211989110158401015840) (16)
then we have
|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)| ge 0 (17)
Now
|119860 (119911) + (1 minus 120572) 119861 (119911)| = 1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 120573) 119896119890119894120579 + (2 minus 120572) 119911minus infinsum119899=2
[(120573Ω119899 minus Π119899) 119896119890119894120579 minus (1 minus 120572)Ω119899 minus Π119899] 1198861198991199111198991003816100381610038161003816100381610038161003816100381610038161003816ge minus ((1 minus 120573) 119896 + (2 minus 120572)) |119911|minus infinsum119899=2
[(120573Ω119899 minus Π119899) 119896 + (1 minus 120572)Ω119899 + Π119899] 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899
(18)
and also
|119860 (119911) minus (1 + 120572) 119861 (119911)| = 1003816100381610038161003816100381610038161003816100381610038161003816((1 minus 120573) 119896119890119894120579 minus 120572) 119911+ infinsum119899=2
[(Π119899 minus 120573Ω119899) 119896119890119894120579 minus (1 + 120572)Ω119899 + Π119899] 1198861198991199111198991003816100381610038161003816100381610038161003816100381610038161003816le ((1 minus 120573) 119896 + 120572) |119911|+ infinsum119899=2
[(Π119899 minus 120573Ω119899) 119896 minus (1 + 120572)Ω119899 + Π119899] 10038161003816100381610038161198861198991003816100381610038161003816 10038161003816100381610038161199111198991003816100381610038161003816
(19)
From (18) and (19) we have
|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)|ge 2 [(1 minus 120572) minus 119896 (1 minus 120573)] |119911|minus 2infinsum119899=2
[(Π119899 minus 120573Ω119899) 119896 + (Π119899 minus 120572Ω119899)] 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899= 2 [[(1 minus 120572) minus 119896 (1 minus 120573)] |119911|minus infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899]
(20)
The last expression is bounded below by 0 ifinfinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(21)
which completes the proof
In the next theorem we prove that condition (11) is alsonecessary for function 119891 isin 119896 minusU(120572 120573 120582 120575)Theorem 3 Let 119891(119911) be given by (1) and in 119881120578 then 119891 isin 119896 minusVU120578(120572 120573 120582 120575) if and only if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(22)
Proof From Theorem 2 we need only to show that 119891 isin119896 minus VU120578(120572 120573 120582 120575) satisfies inequality (22) If 119891 isin 119896 minusVU120578(120572 120573 120582 120575) then by definition we have
Re((1 minus 120572) + suminfin119899=2 (Π119899 minus 120572Ω119899) 119886119899119911119899minus11 + suminfin119899=2Ω119899119886119899119911119899minus1 )ge 119896 100381610038161003816100381610038161003816100381610038161003816
(1 minus 120573) + suminfin119899=2 (Π119899 minus 120573Ω119899) 119886119899119911119899minus11 + suminfin119899=2Ω119899119886119899119911119899minus1100381610038161003816100381610038161003816100381610038161003816
(23)
Since119891 is function of form (1)with the argument propertygiven in class 119881120578 and letting 119911 = 119903119890120579 in the above inequalitywe have
(1 minus 120572) minus suminfin119899=2 (Π119899 minus 120572Ω119899) 10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus11 minus suminfin119899=2Ω119899 10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus1ge 119896((1 minus 120573) minus suminfin119899=2 (Π119899 minus 120573Ω119899) 10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus11 minus suminfin119899=2Ω119899 10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus1 )
(24)
for 119903 rarr 1 and (24) leads to require inequalityinfinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(25)
4 Journal of Function Spaces
The function
119891119899120578 (119911) = 119911 minus (1 minus 120572 minus 119896 (1 minus 120573)) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 1199111198990 le 120578 le 2120587 119899 ge 2
(26)
is extremal function
Corollary 4 Let 119891(119911) given in (1) be in class 119896 minusVU120578(120572 120573120582 120575) Then
10038161003816100381610038161198861198991003816100381610038161003816 le 1 minus 120572 minus 119896 (1 minus 120573)Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 119899 ge 2 (27)
Inequality (27) is attained for the function given in (26)
Theorem 5 Let the function 119891(119911) given in (1) be in class 119896 minusVU120578(120572 120573 120582 120575)Then for |119911| lt 119903 = 1
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge 119903 minus 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 11990321003816100381610038161003816119891 (119911)1003816100381610038161003816 le 119903 + 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 1199032
(28)
The results in (28) are attained for the function given in(26) for 119911 = plusmn119903Proof As we know fromTheorem 3
[Π2 (1 + 119896) minus (119896120573 + 120572)Ω2] infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816le infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(29)
As
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge |119911| minus infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899 ge 119903 minus 1199032 infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816ge 119903 minus 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 1199032
(30)
similarly
1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| + infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899 le 119903 + 1199032 infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816le 119903 + 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 1199032
(31)
This completes the proof
Theorem 6 Let the function 119891(119911) given in (1) be in class 119896 minusVU120578(120572 120573 120582 120575)Then for |119911| lt 119903 = 1
1 minus 2 (1 minus 120572 minus 119896 (1 minus 120573))Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 119903 le 1003816100381610038161003816119891 (119911)1003816100381610038161003816le 1 + 2 (1 minus 120572 minus 119896 (1 minus 120573))Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 119903
(32)
Proof For 119891(119911) given by (1) we have
100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 ge 1 minus infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 ge 1 minus 119903infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le 1 + infinsum
119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 le 1 + 119903infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 (33)
In view of Theorem 3[Π2 (1 + 119896) minus (119896120573 + 120572)Ω2]2infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816le infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(34)
or equivalentlyinfinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 le 2 (1 minus 120572 minus 119896 (1 minus 120573))[Π2 (1 + 119896) minus (119896120573 + 120572)Ω2] (35)
A substitution from (35) into (33) yields inequality (32)which is required
Theorem 7 Let 119891 isin 119896 minus VU120578(120572 120573 120582 120575) with argumentproperty as in class 119881120578 Define 119891119895(119911) = 119911 and
119891119899120578 = 119911 minus 1 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]119911119899 (36)
where 0 le 120578 le 2120587 119899 ge 2Then function 119891(119911) is in class 119896 minus VU120578(120572 120573 120582 120575) if and
only if it can be expressed as
119891 (119911) = infinsum119899=1
120583119899119891119899120578 (37)
where 120583119899 ge 0 (119899 ge 1) and suminfin119899=1 120583119899 = 1Proof Assume that
119891 (119911) = 12058311198911 (119911)+ infinsum119899=2
120583119899 [119911 minus 1 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 119911119899]= infinsum119899=1
120583119899119911minus infinsum119899=2
[ 1 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]120583119899119911119899
(38)
Journal of Function Spaces 5
Then it follows thatinfinsum119899=2
10038161003816100381610038161003816100381610038161003816100381610038161 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899
1003816100381610038161003816100381610038161003816100381610038161003816sdot 120583119899 [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]= infinsum119899=2
120583119899 [1 minus 120572 minus 119896 (1 minus 120573)] le (1 minus 1205831)sdot [1 minus 120572 minus 119896 (1 minus 120573)] le 1 minus 120572 minus 119896 (1 minus 120573)
(39)
by Theorem 3 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Conversely assumethat the function 119891(119911) defined by (1) belongs to class 119896 minusVU120578(120572 120573 120582 120575) and then
10038161003816100381610038161198861198991003816100381610038161003816 le 1 minus 120572 minus 119896 (1 minus 120573)Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 119899 ge 2 (40)
Set
120583119899 = Π119899 (1 + 119896) minus (119896120573 + 120572)Ω1198991 minus 120572 minus 119896 (1 minus 120573) 10038161003816100381610038161198861198991003816100381610038161003816 119899 ge 2 (41)
and 1205831 = 1minussuminfin119899=2 120583119899 119899 ge 2Then119891(119911) = suminfin119899=1 120583119899119891119899120578 and thiscompletes the proof
Theorem 8 Let 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Then 119891(119911) is closeto convex of order 120590(0 le 120590 lt 1) in the disc |119911| lt 1199031 where1199031 = inf [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)119899 (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(42)
Proof As 119891 isin 119881120578 where 119891 is close to convex of order 120590 wehave 100381610038161003816100381610038161198911015840 (119911) minus 110038161003816100381610038161003816 lt 1 minus 120590 (43)
as
100381610038161003816100381610038161198911015840 (119911) minus 110038161003816100381610038161003816 le infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 (44)
this expression is less than 1 minus 120590 ifinfinsum119899=2
1198991 minus 120590 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 lt 1 (45)
By the fact that 119891 isin 119896 minusVU120578(120572 120573 120582 120575) if and only if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 119886119899 le 1 (46)
inequality (43) is true if
1198991 minus 120590119911119899minus1 le Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) (47)
or equivalently
|119911|119899minus1 = [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)119899 (1 minus 120572 minus 119896 (1 minus 120573)) ] (48)
Theorem 9 Let 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Then 119891(119911) is closeto convex of order 120590 (0 le 120590 lt 1) in the disc|119911| lt 1199032 where1199032 = inf [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(49)
Proof As 119891 isin 119881120578 and 119891 is starlike of order 120590 then we have
100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1
100381610038161003816100381610038161003816100381610038161003816 lt 1 minus 120590 (50)
as 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1
100381610038161003816100381610038161003816100381610038161003816 lesuminfin119899=2 (119899 minus 1) 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus11 minus suminfin119899=2 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 (51)
The last expression is less than 1 minus 120590 ifinfinsum119899=2
119899 minus 1205901 minus 120590 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 lt 1 (52)
Using the fact that 119891 isin 119896 minusVU120578(120572 120573 120582 120575) if and only if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 119886119899 le 1 (53)
(50) is true if
119899 minus 1205901 minus 120590 |119911|119899minus1 lt Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) (54)
Or equivalently
|119911|119899minus1 = (1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) (55)
which is required
Theorem 10 Let119891 isin 119896minusVU120578(120572 120573 120582 120575)Then119891(119911) is convexof order 120590 (0 le 120590 lt 1) in the disc|119911| lt 1199033 where1199033 = inf [(1 minus 120590) [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]119899 (119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(56)
Proof Using the fact that 119891 is convex if and only if 1199111198911015840is starlike following the lines of Theorem 9 we have therequired results
Theorem 11 Let 119891119895(119911) (119895 = 1 2 ) given by (6) be in class119896 minusVU120578(120572 120573 120582 120575) Then (1198911 lowast 1198912) isin 119896 minusVU120578(1206011 120582 120575) for
6 Journal of Function Spaces
1206011 = (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus (Π119899 (1 + 119896) minus 119896120573Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))2(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus Ω119899 (1 minus 120572 minus 119896 (1 minus 120573))2 (57)
Proof We need to prove the largest 1206011 such that
(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(1 minus 1206011 minus 119896 (1 minus 120573)) 11988611989911198861198992 le 1 (58)
FromTheorem 3 we haveinfinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 1198861198991 le 1infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 1198861198992 le 1(59)
By Cauchy-Schwarz inequality we have
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ]radic11988611989911198861198992 le 1 (60)
Thus it is sufficient to show
[Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899(1 minus 1206011 minus 119896 (1 minus 120573)) ] 11988611989911198861198992le [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ]radic11988611989911198861198992 119899 ge 2
(61)
For 119899 ge 2radic11988611989911198861198992le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (1 minus 1206011 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))
(62)
Note that
radic11988611989911198861198992 le (1 minus 120572 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (63)
We need to show
(1 minus 120572 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (1 minus 1206011 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))
(64)
or equivalently
1206011 le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus (Π119899 (1 + 119896) minus 119896120573Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))2(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus Ω119899 (1 minus 120572 minus 119896 (1 minus 120573))2 = 120596 (119899) (65)
120596(119899) is an increasing function for 119899 ge 2 For 119899 = 2 in (65)
1206011 le 120596 (2) = (Π2 (1 + 119896) minus (119896120573 + 120572)Ω2)2 minus (Π2 (1 + 119896) minus 119896120573Ω2) (1 minus 120572 minus 119896 (1 minus 120573))2(Π2 (1 + 119896) minus (119896120573 + 120572)Ω2)2 minus Ω2 (1 minus 120572 minus 119896 (1 minus 120573))2 (66)
which proves main assertion of Theorem 11
Conflicts of Interest
The authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors jointly work on the results and they read andapproved the final manuscript
Acknowledgments
The work here is supported by MOHE Grant FRGS12016STG06UKM011
References
[1] A W Goodman ldquoOn uniformly convex functionsrdquo AnnalesPolonici Mathematici vol 56 no 1 pp 87ndash92 1991
[2] A W Goodman ldquoOn uniformly starlike functionsrdquo Journalof Mathematical Analysis and Applications vol 155 no 2 pp364ndash370 1991
Journal of Function Spaces 7
[3] W CMa andDMinda ldquoUniformly convex functionsrdquoAnnalesPolonici Mathematici vol 57 no 2 pp 165ndash175 1992
[4] F Roslashnning ldquoUniformly convex functions and a correspondingclass of starlike functionsrdquo Proceedings of the American Mathe-matical Society vol 118 no 1 pp 189ndash196 1993
[5] J Sokol and A Wisniowska-Wajnryb ldquoOn some classes ofstarlike functions related with parabolardquo Folia Sci Univ TechResov vol 121 no 18 pp 35ndash42 1993
[6] J Sokol and A Wisniowska-Wajnryb ldquoOn certain problem inthe classes of k-starlike functionsrdquo Computers amp Mathematicswith Applications vol 62 no 12 pp 4733ndash4741 2011
[7] S Kanas and A Wisniowska ldquoConic regions and k-uniformconvexityrdquo Journal of Computational and Applied Mathematicsvol 105 no 1-2 pp 327ndash336 1999
[8] S Kanas andHM Srivastava ldquoLinear operators associatedwithk-uniformly convex functionsrdquo Integral Transforms and SpecialFunction vol 9 no 2 pp 121ndash132 2000
[9] AMannino ldquoSome inequalities concerning starlike and convexfunctionsrdquo General Mathematics vol 12 no 1 pp 5ndash12 2004
[10] S Ponnusamy and M Vuorinen ldquoUnivalence and convexityproperties for Gaussian hypergeometric functionsrdquo The RockyMountain Journal of Mathematics vol 31 no 1 pp 327ndash3532001
[11] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975
[12] F Ronning ldquoIntegral representation for bounded starlike func-tionsrdquoAnnales Polonici Mathematici vol 60 no 3 pp 289ndash2971995
[13] R M El-Ashwah M K Aouf A A Hassan and A H HassanldquoCertain new classes of analytic functions with varying argu-mentsrdquo Journal of Complex Analysis vol 2013 Article ID958210 5 pages 2013
[14] RM Ali S R Mondal and V Ravichandran ldquoOn the Janowskiconvexity and starlikeness of the confluent hypergeometricfunctionrdquo Bulletin of the Belgian Mathematical Society SimonStevin vol 22 no 2 pp 227ndash250 2015
[15] R M Ali V Ravichandran and N Seenivasagan ldquoSubordina-tion and superordination of the Liu-Srivastava linear operatoron meromorphic functionsrdquo Bulletin of the Malaysian Mathe-matical Sciences Society vol 31 no 2 pp 193ndash207 2008
[16] R M Ali and V Ravichandran ldquoUniformly convex and uni-formly starlike functionsrdquo Mathematics Newsletter vol 21 pp16ndash30 2011
[17] S Altınkaya and S Yalcın ldquoCoefficient estimates for two newsubclasses of bi-univalent functions with respect to symmetricpointsrdquo Journal of Function Spaces Article ID 145242 2014
[18] M K Aouf H M Hossen and A Y Lashin ldquoOn certain fam-ilies of analytic functions with negative coefficientsrdquo IndianJournal of Pure and AppliedMathematics vol 31 no 8 pp 999ndash1015 2000
[19] M K Aouf A A Shamandy A O Mostafa and A K WagdyldquoCertain subclasses of uniformly starlike and convex functionsdefined by convolution with negative coefficientsrdquo Matem-atichki Vesnik vol 65 no 1 pp 14ndash28 2013
[20] A Kaminski and S Mincheva-Kaminska ldquoCompatibility con-ditions and the convolution of functions and generalized func-tionsrdquo Journal of Function Spaces and Applications vol 2013Article ID 356724 11 pages 2013
[21] N Magesh ldquoCertain subclasses of uniformly convex functionsof order 120572 and type 120573 with varying argumentsrdquo Journal of theEgyptian Mathematical Society vol 21 no 3 pp 184ndash189 2013
[22] K I Noor ldquoSome properties of certain analytic functionsrdquoJournal of Natural Geometry vol 7 no 1 pp 11ndash20 1995
[23] R Bharati R Parvatham and A Swaminathan ldquoOn subclassesof uniformly convex functions and corresponding class of star-like functionsrdquo Tamkang Journal of Mathematics vol 28 no 1pp 17ndash32 1997
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Stochastic AnalysisInternational Journal of
2 Journal of Function Spaces
A lot of authors obtain very useful properties of UCVand UST and their generalization in several direction forexample see [1 2 7 8 10 11] and reference cited therein
For (0 le 120572 lt 1) in [4] (see also [12]) Ronning introducedthe following two important subclasses 119896 minusUST(120572) and 119896 minusUCV(120572) as
119891 isin 119896 minusUST (120572) lArrrArrRe1199111198911015840 (119911)119891 (119911) minus 120572 gt 119896
100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1
100381610038161003816100381610038161003816100381610038161003816 119911 isin Δ
119891 isin 119896 minusUCV (120572) lArrrArr1199111198911015840 isin 119896 minusUST (120572)
(4)
Recently in [13] El-Ashwah et al introduced two impor-tant subclass 119896 minus UCV(120572 120573) and 119896 minus UST(120572 120573) of 119896-uniformly convex starlike functions as
119891 isin 119896 minusUCV (120572 120573) lArrrArrRe(1199111198911015840 (119911))10158401198911015840 (119911) minus 120572 gt 119896
100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1199111198911015840 (119911))10158401198911015840 (119911) minus 120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816
119911 isin Δ119891 isin 119896 minusUST (120572 120573) lArrrArr
Re1199111198911015840 (119911)119891 (119911) minus 120572 gt 119896100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 120573
100381610038161003816100381610038161003816100381610038161003816 119911 isin Δ
(5)
where (0 le 120572 lt 120573 le 1) and 119896(1 minus 120573) lt 1 minus 120572Let 119891119895 (119895 = 1 2 ) be defined by
119891119895 (119911) = 119911 + infinsum119899=2
119886119899119895119911119899 119886119899119895 ge 0 119895 isin 119873 (6)
then the modified Hadmard product of 1198911(119911) and 1198912(119911) isdefined by
(1198911 lowast 1198912) (119911) = 119911 minus infinsum119899=2
11988611989911198861198992119911119899 (7)
We denote T by subclass of S consisting of functionshaving all negative coefficients in their Maclaurinrsquos seriesexpansions so any 119891 isin T has a series of the form
119891 (119911) = 119911 minus infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816 119911119899 119911 isin Δ (8)
Let V120578 be the class of functions 119891 isin S given in (1) forwhich arg(119886119899) = 120587 + (119899 minus 1)120578 119899 ge 2 Note thatV0 = T [11]
In recent years more and more researchers are interestedin the above defined classes (see [9 11 14ndash22])
In this paper by taking inspiration from the abovecited paper we introduce some new subclasses of analyticfunctions and obtain some interesting results
Definition 1 For (0 le 120572 lt 120573 le 1) 0 le 120575 lt 1 119896(1minus120573) lt 1minus120572and 0 le 120582 lt 1 a function 119891 isin S is in class 119896 minusU(120572 120573 120582 120575) ifand only if
Re(1 minus 120575) 1199111198911015840 + 120575 (1199111198911015840 + (1 + 2120582) 119911211989110158401015840 + 1205821199113119891101584010158401015840)(1 minus 120575) 119891 + 120575 (1199111198911015840 + 120582119911211989110158401015840)minus 120572ge 119896 10038161003816100381610038161003816100381610038161003816100381610038161003816
(1 minus 120575) 1199111198911015840 + 120575 (1199111198911015840 + (1 + 2120582) 119911211989110158401015840 + 1205821199113119891101584010158401015840)(1 minus 120575) 119891 + 120575 (1199111198911015840 + 120582119911211989110158401015840)minus 12057310038161003816100381610038161003816100381610038161003816100381610038161003816 119911 isin Δ
(9)
Also
119896 minusVU120578 (120572 120573 120582 120575) = 119896 minusU (120572 120573 120582 120575) capV120578 (10)
It is worth mentioning that for special values of parame-ters these classes were extensively studied by many authorshere we mention few of them
(1) 119896 minusU(120572 120573 120582 1) = 119896 minus 119880(120582 120573 120572) [21](2) 119896 minusVU0(120572 120573 120582 1) = 119896 minus 119881119880120578(120582 120573 120572) [21](3) 0 minusVU0(120572 1 0 1) = CV(120572) [11](4) 119896 minusVU0(120572 1 0 1) = 119896 minusUCV(120572) [23](5) 1 minusU(120572 1 0 1) = UCV(120572) [4](6) 119896 minusU(120572 120573 0 0) = 119896 minusUST(120572 120573) [13]Throughout the paper minus1 le 120572 le 120573 le 1 0 le 120582 lt 11 minus 120572 gt 119896(1 minus 120573) and 119911 isin Δ unless otherwise stated
2 Main Results
Theorem 2 A function 119891(119911) given by (1) is in class 119896 minusU(120572 120573 120582 120575) if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(11)
where
Π119899 = (119899 + 120575119899 (119899 minus 1) (1 + 120582119899)) Ω119899 = 1 minus 120575 + 120575119899 (1 + 120582 (119899 minus 1)) (12)
Proof It is sufficient to prove that inequality (9) holds As weknow
Re (119908) gt 119896 1003816100381610038161003816119908 minus 1205731003816100381610038161003816 + 120572iff Re ((1 + 119896119890119894120579)119908 minus 120573119896119890119894120579) ge 120572 (13)
Journal of Function Spaces 3
then inequality (9) can be written as
Re((1 + 119896119890119894120579)sdot (1 minus 120575) 1199111198911015840 + 120575 (1199111198911015840 + (1 + 2120582) 119911211989110158401015840 + 1205821199113119891101584010158401015840)(1 minus 120575) 119891 + 120575 (1199111198911015840 + 120582119911211989110158401015840)minus 120573119896119890119894120579) ge 120572
(14)
This is
Re(119860 (119911)119861 (119911) ) ge 120572 (15)
where
119860 (119911)= (1 + 119896119890119894120579) (1199111198911015840 + 120575 ((1 + 2120582) 119911211989110158401015840 + 1205821199113119891101584010158401015840))minus 120573119896119890119894120579 ((1 minus 120575) 119891 + 120575 (1199111198911015840 + 120582119911211989110158401015840))
119861 (119911) = (1 minus 120575) 119891 + 120575 (1199111198911015840 + 120582119911211989110158401015840) (16)
then we have
|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)| ge 0 (17)
Now
|119860 (119911) + (1 minus 120572) 119861 (119911)| = 1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 120573) 119896119890119894120579 + (2 minus 120572) 119911minus infinsum119899=2
[(120573Ω119899 minus Π119899) 119896119890119894120579 minus (1 minus 120572)Ω119899 minus Π119899] 1198861198991199111198991003816100381610038161003816100381610038161003816100381610038161003816ge minus ((1 minus 120573) 119896 + (2 minus 120572)) |119911|minus infinsum119899=2
[(120573Ω119899 minus Π119899) 119896 + (1 minus 120572)Ω119899 + Π119899] 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899
(18)
and also
|119860 (119911) minus (1 + 120572) 119861 (119911)| = 1003816100381610038161003816100381610038161003816100381610038161003816((1 minus 120573) 119896119890119894120579 minus 120572) 119911+ infinsum119899=2
[(Π119899 minus 120573Ω119899) 119896119890119894120579 minus (1 + 120572)Ω119899 + Π119899] 1198861198991199111198991003816100381610038161003816100381610038161003816100381610038161003816le ((1 minus 120573) 119896 + 120572) |119911|+ infinsum119899=2
[(Π119899 minus 120573Ω119899) 119896 minus (1 + 120572)Ω119899 + Π119899] 10038161003816100381610038161198861198991003816100381610038161003816 10038161003816100381610038161199111198991003816100381610038161003816
(19)
From (18) and (19) we have
|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)|ge 2 [(1 minus 120572) minus 119896 (1 minus 120573)] |119911|minus 2infinsum119899=2
[(Π119899 minus 120573Ω119899) 119896 + (Π119899 minus 120572Ω119899)] 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899= 2 [[(1 minus 120572) minus 119896 (1 minus 120573)] |119911|minus infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899]
(20)
The last expression is bounded below by 0 ifinfinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(21)
which completes the proof
In the next theorem we prove that condition (11) is alsonecessary for function 119891 isin 119896 minusU(120572 120573 120582 120575)Theorem 3 Let 119891(119911) be given by (1) and in 119881120578 then 119891 isin 119896 minusVU120578(120572 120573 120582 120575) if and only if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(22)
Proof From Theorem 2 we need only to show that 119891 isin119896 minus VU120578(120572 120573 120582 120575) satisfies inequality (22) If 119891 isin 119896 minusVU120578(120572 120573 120582 120575) then by definition we have
Re((1 minus 120572) + suminfin119899=2 (Π119899 minus 120572Ω119899) 119886119899119911119899minus11 + suminfin119899=2Ω119899119886119899119911119899minus1 )ge 119896 100381610038161003816100381610038161003816100381610038161003816
(1 minus 120573) + suminfin119899=2 (Π119899 minus 120573Ω119899) 119886119899119911119899minus11 + suminfin119899=2Ω119899119886119899119911119899minus1100381610038161003816100381610038161003816100381610038161003816
(23)
Since119891 is function of form (1)with the argument propertygiven in class 119881120578 and letting 119911 = 119903119890120579 in the above inequalitywe have
(1 minus 120572) minus suminfin119899=2 (Π119899 minus 120572Ω119899) 10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus11 minus suminfin119899=2Ω119899 10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus1ge 119896((1 minus 120573) minus suminfin119899=2 (Π119899 minus 120573Ω119899) 10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus11 minus suminfin119899=2Ω119899 10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus1 )
(24)
for 119903 rarr 1 and (24) leads to require inequalityinfinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(25)
4 Journal of Function Spaces
The function
119891119899120578 (119911) = 119911 minus (1 minus 120572 minus 119896 (1 minus 120573)) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 1199111198990 le 120578 le 2120587 119899 ge 2
(26)
is extremal function
Corollary 4 Let 119891(119911) given in (1) be in class 119896 minusVU120578(120572 120573120582 120575) Then
10038161003816100381610038161198861198991003816100381610038161003816 le 1 minus 120572 minus 119896 (1 minus 120573)Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 119899 ge 2 (27)
Inequality (27) is attained for the function given in (26)
Theorem 5 Let the function 119891(119911) given in (1) be in class 119896 minusVU120578(120572 120573 120582 120575)Then for |119911| lt 119903 = 1
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge 119903 minus 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 11990321003816100381610038161003816119891 (119911)1003816100381610038161003816 le 119903 + 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 1199032
(28)
The results in (28) are attained for the function given in(26) for 119911 = plusmn119903Proof As we know fromTheorem 3
[Π2 (1 + 119896) minus (119896120573 + 120572)Ω2] infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816le infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(29)
As
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge |119911| minus infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899 ge 119903 minus 1199032 infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816ge 119903 minus 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 1199032
(30)
similarly
1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| + infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899 le 119903 + 1199032 infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816le 119903 + 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 1199032
(31)
This completes the proof
Theorem 6 Let the function 119891(119911) given in (1) be in class 119896 minusVU120578(120572 120573 120582 120575)Then for |119911| lt 119903 = 1
1 minus 2 (1 minus 120572 minus 119896 (1 minus 120573))Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 119903 le 1003816100381610038161003816119891 (119911)1003816100381610038161003816le 1 + 2 (1 minus 120572 minus 119896 (1 minus 120573))Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 119903
(32)
Proof For 119891(119911) given by (1) we have
100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 ge 1 minus infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 ge 1 minus 119903infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le 1 + infinsum
119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 le 1 + 119903infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 (33)
In view of Theorem 3[Π2 (1 + 119896) minus (119896120573 + 120572)Ω2]2infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816le infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(34)
or equivalentlyinfinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 le 2 (1 minus 120572 minus 119896 (1 minus 120573))[Π2 (1 + 119896) minus (119896120573 + 120572)Ω2] (35)
A substitution from (35) into (33) yields inequality (32)which is required
Theorem 7 Let 119891 isin 119896 minus VU120578(120572 120573 120582 120575) with argumentproperty as in class 119881120578 Define 119891119895(119911) = 119911 and
119891119899120578 = 119911 minus 1 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]119911119899 (36)
where 0 le 120578 le 2120587 119899 ge 2Then function 119891(119911) is in class 119896 minus VU120578(120572 120573 120582 120575) if and
only if it can be expressed as
119891 (119911) = infinsum119899=1
120583119899119891119899120578 (37)
where 120583119899 ge 0 (119899 ge 1) and suminfin119899=1 120583119899 = 1Proof Assume that
119891 (119911) = 12058311198911 (119911)+ infinsum119899=2
120583119899 [119911 minus 1 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 119911119899]= infinsum119899=1
120583119899119911minus infinsum119899=2
[ 1 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]120583119899119911119899
(38)
Journal of Function Spaces 5
Then it follows thatinfinsum119899=2
10038161003816100381610038161003816100381610038161003816100381610038161 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899
1003816100381610038161003816100381610038161003816100381610038161003816sdot 120583119899 [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]= infinsum119899=2
120583119899 [1 minus 120572 minus 119896 (1 minus 120573)] le (1 minus 1205831)sdot [1 minus 120572 minus 119896 (1 minus 120573)] le 1 minus 120572 minus 119896 (1 minus 120573)
(39)
by Theorem 3 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Conversely assumethat the function 119891(119911) defined by (1) belongs to class 119896 minusVU120578(120572 120573 120582 120575) and then
10038161003816100381610038161198861198991003816100381610038161003816 le 1 minus 120572 minus 119896 (1 minus 120573)Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 119899 ge 2 (40)
Set
120583119899 = Π119899 (1 + 119896) minus (119896120573 + 120572)Ω1198991 minus 120572 minus 119896 (1 minus 120573) 10038161003816100381610038161198861198991003816100381610038161003816 119899 ge 2 (41)
and 1205831 = 1minussuminfin119899=2 120583119899 119899 ge 2Then119891(119911) = suminfin119899=1 120583119899119891119899120578 and thiscompletes the proof
Theorem 8 Let 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Then 119891(119911) is closeto convex of order 120590(0 le 120590 lt 1) in the disc |119911| lt 1199031 where1199031 = inf [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)119899 (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(42)
Proof As 119891 isin 119881120578 where 119891 is close to convex of order 120590 wehave 100381610038161003816100381610038161198911015840 (119911) minus 110038161003816100381610038161003816 lt 1 minus 120590 (43)
as
100381610038161003816100381610038161198911015840 (119911) minus 110038161003816100381610038161003816 le infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 (44)
this expression is less than 1 minus 120590 ifinfinsum119899=2
1198991 minus 120590 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 lt 1 (45)
By the fact that 119891 isin 119896 minusVU120578(120572 120573 120582 120575) if and only if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 119886119899 le 1 (46)
inequality (43) is true if
1198991 minus 120590119911119899minus1 le Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) (47)
or equivalently
|119911|119899minus1 = [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)119899 (1 minus 120572 minus 119896 (1 minus 120573)) ] (48)
Theorem 9 Let 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Then 119891(119911) is closeto convex of order 120590 (0 le 120590 lt 1) in the disc|119911| lt 1199032 where1199032 = inf [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(49)
Proof As 119891 isin 119881120578 and 119891 is starlike of order 120590 then we have
100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1
100381610038161003816100381610038161003816100381610038161003816 lt 1 minus 120590 (50)
as 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1
100381610038161003816100381610038161003816100381610038161003816 lesuminfin119899=2 (119899 minus 1) 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus11 minus suminfin119899=2 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 (51)
The last expression is less than 1 minus 120590 ifinfinsum119899=2
119899 minus 1205901 minus 120590 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 lt 1 (52)
Using the fact that 119891 isin 119896 minusVU120578(120572 120573 120582 120575) if and only if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 119886119899 le 1 (53)
(50) is true if
119899 minus 1205901 minus 120590 |119911|119899minus1 lt Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) (54)
Or equivalently
|119911|119899minus1 = (1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) (55)
which is required
Theorem 10 Let119891 isin 119896minusVU120578(120572 120573 120582 120575)Then119891(119911) is convexof order 120590 (0 le 120590 lt 1) in the disc|119911| lt 1199033 where1199033 = inf [(1 minus 120590) [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]119899 (119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(56)
Proof Using the fact that 119891 is convex if and only if 1199111198911015840is starlike following the lines of Theorem 9 we have therequired results
Theorem 11 Let 119891119895(119911) (119895 = 1 2 ) given by (6) be in class119896 minusVU120578(120572 120573 120582 120575) Then (1198911 lowast 1198912) isin 119896 minusVU120578(1206011 120582 120575) for
6 Journal of Function Spaces
1206011 = (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus (Π119899 (1 + 119896) minus 119896120573Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))2(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus Ω119899 (1 minus 120572 minus 119896 (1 minus 120573))2 (57)
Proof We need to prove the largest 1206011 such that
(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(1 minus 1206011 minus 119896 (1 minus 120573)) 11988611989911198861198992 le 1 (58)
FromTheorem 3 we haveinfinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 1198861198991 le 1infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 1198861198992 le 1(59)
By Cauchy-Schwarz inequality we have
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ]radic11988611989911198861198992 le 1 (60)
Thus it is sufficient to show
[Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899(1 minus 1206011 minus 119896 (1 minus 120573)) ] 11988611989911198861198992le [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ]radic11988611989911198861198992 119899 ge 2
(61)
For 119899 ge 2radic11988611989911198861198992le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (1 minus 1206011 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))
(62)
Note that
radic11988611989911198861198992 le (1 minus 120572 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (63)
We need to show
(1 minus 120572 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (1 minus 1206011 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))
(64)
or equivalently
1206011 le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus (Π119899 (1 + 119896) minus 119896120573Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))2(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus Ω119899 (1 minus 120572 minus 119896 (1 minus 120573))2 = 120596 (119899) (65)
120596(119899) is an increasing function for 119899 ge 2 For 119899 = 2 in (65)
1206011 le 120596 (2) = (Π2 (1 + 119896) minus (119896120573 + 120572)Ω2)2 minus (Π2 (1 + 119896) minus 119896120573Ω2) (1 minus 120572 minus 119896 (1 minus 120573))2(Π2 (1 + 119896) minus (119896120573 + 120572)Ω2)2 minus Ω2 (1 minus 120572 minus 119896 (1 minus 120573))2 (66)
which proves main assertion of Theorem 11
Conflicts of Interest
The authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors jointly work on the results and they read andapproved the final manuscript
Acknowledgments
The work here is supported by MOHE Grant FRGS12016STG06UKM011
References
[1] A W Goodman ldquoOn uniformly convex functionsrdquo AnnalesPolonici Mathematici vol 56 no 1 pp 87ndash92 1991
[2] A W Goodman ldquoOn uniformly starlike functionsrdquo Journalof Mathematical Analysis and Applications vol 155 no 2 pp364ndash370 1991
Journal of Function Spaces 7
[3] W CMa andDMinda ldquoUniformly convex functionsrdquoAnnalesPolonici Mathematici vol 57 no 2 pp 165ndash175 1992
[4] F Roslashnning ldquoUniformly convex functions and a correspondingclass of starlike functionsrdquo Proceedings of the American Mathe-matical Society vol 118 no 1 pp 189ndash196 1993
[5] J Sokol and A Wisniowska-Wajnryb ldquoOn some classes ofstarlike functions related with parabolardquo Folia Sci Univ TechResov vol 121 no 18 pp 35ndash42 1993
[6] J Sokol and A Wisniowska-Wajnryb ldquoOn certain problem inthe classes of k-starlike functionsrdquo Computers amp Mathematicswith Applications vol 62 no 12 pp 4733ndash4741 2011
[7] S Kanas and A Wisniowska ldquoConic regions and k-uniformconvexityrdquo Journal of Computational and Applied Mathematicsvol 105 no 1-2 pp 327ndash336 1999
[8] S Kanas andHM Srivastava ldquoLinear operators associatedwithk-uniformly convex functionsrdquo Integral Transforms and SpecialFunction vol 9 no 2 pp 121ndash132 2000
[9] AMannino ldquoSome inequalities concerning starlike and convexfunctionsrdquo General Mathematics vol 12 no 1 pp 5ndash12 2004
[10] S Ponnusamy and M Vuorinen ldquoUnivalence and convexityproperties for Gaussian hypergeometric functionsrdquo The RockyMountain Journal of Mathematics vol 31 no 1 pp 327ndash3532001
[11] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975
[12] F Ronning ldquoIntegral representation for bounded starlike func-tionsrdquoAnnales Polonici Mathematici vol 60 no 3 pp 289ndash2971995
[13] R M El-Ashwah M K Aouf A A Hassan and A H HassanldquoCertain new classes of analytic functions with varying argu-mentsrdquo Journal of Complex Analysis vol 2013 Article ID958210 5 pages 2013
[14] RM Ali S R Mondal and V Ravichandran ldquoOn the Janowskiconvexity and starlikeness of the confluent hypergeometricfunctionrdquo Bulletin of the Belgian Mathematical Society SimonStevin vol 22 no 2 pp 227ndash250 2015
[15] R M Ali V Ravichandran and N Seenivasagan ldquoSubordina-tion and superordination of the Liu-Srivastava linear operatoron meromorphic functionsrdquo Bulletin of the Malaysian Mathe-matical Sciences Society vol 31 no 2 pp 193ndash207 2008
[16] R M Ali and V Ravichandran ldquoUniformly convex and uni-formly starlike functionsrdquo Mathematics Newsletter vol 21 pp16ndash30 2011
[17] S Altınkaya and S Yalcın ldquoCoefficient estimates for two newsubclasses of bi-univalent functions with respect to symmetricpointsrdquo Journal of Function Spaces Article ID 145242 2014
[18] M K Aouf H M Hossen and A Y Lashin ldquoOn certain fam-ilies of analytic functions with negative coefficientsrdquo IndianJournal of Pure and AppliedMathematics vol 31 no 8 pp 999ndash1015 2000
[19] M K Aouf A A Shamandy A O Mostafa and A K WagdyldquoCertain subclasses of uniformly starlike and convex functionsdefined by convolution with negative coefficientsrdquo Matem-atichki Vesnik vol 65 no 1 pp 14ndash28 2013
[20] A Kaminski and S Mincheva-Kaminska ldquoCompatibility con-ditions and the convolution of functions and generalized func-tionsrdquo Journal of Function Spaces and Applications vol 2013Article ID 356724 11 pages 2013
[21] N Magesh ldquoCertain subclasses of uniformly convex functionsof order 120572 and type 120573 with varying argumentsrdquo Journal of theEgyptian Mathematical Society vol 21 no 3 pp 184ndash189 2013
[22] K I Noor ldquoSome properties of certain analytic functionsrdquoJournal of Natural Geometry vol 7 no 1 pp 11ndash20 1995
[23] R Bharati R Parvatham and A Swaminathan ldquoOn subclassesof uniformly convex functions and corresponding class of star-like functionsrdquo Tamkang Journal of Mathematics vol 28 no 1pp 17ndash32 1997
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Journal of Function Spaces 3
then inequality (9) can be written as
Re((1 + 119896119890119894120579)sdot (1 minus 120575) 1199111198911015840 + 120575 (1199111198911015840 + (1 + 2120582) 119911211989110158401015840 + 1205821199113119891101584010158401015840)(1 minus 120575) 119891 + 120575 (1199111198911015840 + 120582119911211989110158401015840)minus 120573119896119890119894120579) ge 120572
(14)
This is
Re(119860 (119911)119861 (119911) ) ge 120572 (15)
where
119860 (119911)= (1 + 119896119890119894120579) (1199111198911015840 + 120575 ((1 + 2120582) 119911211989110158401015840 + 1205821199113119891101584010158401015840))minus 120573119896119890119894120579 ((1 minus 120575) 119891 + 120575 (1199111198911015840 + 120582119911211989110158401015840))
119861 (119911) = (1 minus 120575) 119891 + 120575 (1199111198911015840 + 120582119911211989110158401015840) (16)
then we have
|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)| ge 0 (17)
Now
|119860 (119911) + (1 minus 120572) 119861 (119911)| = 1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 120573) 119896119890119894120579 + (2 minus 120572) 119911minus infinsum119899=2
[(120573Ω119899 minus Π119899) 119896119890119894120579 minus (1 minus 120572)Ω119899 minus Π119899] 1198861198991199111198991003816100381610038161003816100381610038161003816100381610038161003816ge minus ((1 minus 120573) 119896 + (2 minus 120572)) |119911|minus infinsum119899=2
[(120573Ω119899 minus Π119899) 119896 + (1 minus 120572)Ω119899 + Π119899] 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899
(18)
and also
|119860 (119911) minus (1 + 120572) 119861 (119911)| = 1003816100381610038161003816100381610038161003816100381610038161003816((1 minus 120573) 119896119890119894120579 minus 120572) 119911+ infinsum119899=2
[(Π119899 minus 120573Ω119899) 119896119890119894120579 minus (1 + 120572)Ω119899 + Π119899] 1198861198991199111198991003816100381610038161003816100381610038161003816100381610038161003816le ((1 minus 120573) 119896 + 120572) |119911|+ infinsum119899=2
[(Π119899 minus 120573Ω119899) 119896 minus (1 + 120572)Ω119899 + Π119899] 10038161003816100381610038161198861198991003816100381610038161003816 10038161003816100381610038161199111198991003816100381610038161003816
(19)
From (18) and (19) we have
|119860 (119911) + (1 minus 120572) 119861 (119911)| minus |119860 (119911) minus (1 + 120572) 119861 (119911)|ge 2 [(1 minus 120572) minus 119896 (1 minus 120573)] |119911|minus 2infinsum119899=2
[(Π119899 minus 120573Ω119899) 119896 + (Π119899 minus 120572Ω119899)] 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899= 2 [[(1 minus 120572) minus 119896 (1 minus 120573)] |119911|minus infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899]
(20)
The last expression is bounded below by 0 ifinfinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(21)
which completes the proof
In the next theorem we prove that condition (11) is alsonecessary for function 119891 isin 119896 minusU(120572 120573 120582 120575)Theorem 3 Let 119891(119911) be given by (1) and in 119881120578 then 119891 isin 119896 minusVU120578(120572 120573 120582 120575) if and only if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(22)
Proof From Theorem 2 we need only to show that 119891 isin119896 minus VU120578(120572 120573 120582 120575) satisfies inequality (22) If 119891 isin 119896 minusVU120578(120572 120573 120582 120575) then by definition we have
Re((1 minus 120572) + suminfin119899=2 (Π119899 minus 120572Ω119899) 119886119899119911119899minus11 + suminfin119899=2Ω119899119886119899119911119899minus1 )ge 119896 100381610038161003816100381610038161003816100381610038161003816
(1 minus 120573) + suminfin119899=2 (Π119899 minus 120573Ω119899) 119886119899119911119899minus11 + suminfin119899=2Ω119899119886119899119911119899minus1100381610038161003816100381610038161003816100381610038161003816
(23)
Since119891 is function of form (1)with the argument propertygiven in class 119881120578 and letting 119911 = 119903119890120579 in the above inequalitywe have
(1 minus 120572) minus suminfin119899=2 (Π119899 minus 120572Ω119899) 10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus11 minus suminfin119899=2Ω119899 10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus1ge 119896((1 minus 120573) minus suminfin119899=2 (Π119899 minus 120573Ω119899) 10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus11 minus suminfin119899=2Ω119899 10038161003816100381610038161198861198991003816100381610038161003816 119903119899minus1 )
(24)
for 119903 rarr 1 and (24) leads to require inequalityinfinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(25)
4 Journal of Function Spaces
The function
119891119899120578 (119911) = 119911 minus (1 minus 120572 minus 119896 (1 minus 120573)) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 1199111198990 le 120578 le 2120587 119899 ge 2
(26)
is extremal function
Corollary 4 Let 119891(119911) given in (1) be in class 119896 minusVU120578(120572 120573120582 120575) Then
10038161003816100381610038161198861198991003816100381610038161003816 le 1 minus 120572 minus 119896 (1 minus 120573)Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 119899 ge 2 (27)
Inequality (27) is attained for the function given in (26)
Theorem 5 Let the function 119891(119911) given in (1) be in class 119896 minusVU120578(120572 120573 120582 120575)Then for |119911| lt 119903 = 1
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge 119903 minus 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 11990321003816100381610038161003816119891 (119911)1003816100381610038161003816 le 119903 + 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 1199032
(28)
The results in (28) are attained for the function given in(26) for 119911 = plusmn119903Proof As we know fromTheorem 3
[Π2 (1 + 119896) minus (119896120573 + 120572)Ω2] infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816le infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(29)
As
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge |119911| minus infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899 ge 119903 minus 1199032 infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816ge 119903 minus 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 1199032
(30)
similarly
1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| + infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899 le 119903 + 1199032 infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816le 119903 + 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 1199032
(31)
This completes the proof
Theorem 6 Let the function 119891(119911) given in (1) be in class 119896 minusVU120578(120572 120573 120582 120575)Then for |119911| lt 119903 = 1
1 minus 2 (1 minus 120572 minus 119896 (1 minus 120573))Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 119903 le 1003816100381610038161003816119891 (119911)1003816100381610038161003816le 1 + 2 (1 minus 120572 minus 119896 (1 minus 120573))Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 119903
(32)
Proof For 119891(119911) given by (1) we have
100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 ge 1 minus infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 ge 1 minus 119903infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le 1 + infinsum
119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 le 1 + 119903infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 (33)
In view of Theorem 3[Π2 (1 + 119896) minus (119896120573 + 120572)Ω2]2infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816le infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(34)
or equivalentlyinfinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 le 2 (1 minus 120572 minus 119896 (1 minus 120573))[Π2 (1 + 119896) minus (119896120573 + 120572)Ω2] (35)
A substitution from (35) into (33) yields inequality (32)which is required
Theorem 7 Let 119891 isin 119896 minus VU120578(120572 120573 120582 120575) with argumentproperty as in class 119881120578 Define 119891119895(119911) = 119911 and
119891119899120578 = 119911 minus 1 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]119911119899 (36)
where 0 le 120578 le 2120587 119899 ge 2Then function 119891(119911) is in class 119896 minus VU120578(120572 120573 120582 120575) if and
only if it can be expressed as
119891 (119911) = infinsum119899=1
120583119899119891119899120578 (37)
where 120583119899 ge 0 (119899 ge 1) and suminfin119899=1 120583119899 = 1Proof Assume that
119891 (119911) = 12058311198911 (119911)+ infinsum119899=2
120583119899 [119911 minus 1 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 119911119899]= infinsum119899=1
120583119899119911minus infinsum119899=2
[ 1 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]120583119899119911119899
(38)
Journal of Function Spaces 5
Then it follows thatinfinsum119899=2
10038161003816100381610038161003816100381610038161003816100381610038161 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899
1003816100381610038161003816100381610038161003816100381610038161003816sdot 120583119899 [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]= infinsum119899=2
120583119899 [1 minus 120572 minus 119896 (1 minus 120573)] le (1 minus 1205831)sdot [1 minus 120572 minus 119896 (1 minus 120573)] le 1 minus 120572 minus 119896 (1 minus 120573)
(39)
by Theorem 3 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Conversely assumethat the function 119891(119911) defined by (1) belongs to class 119896 minusVU120578(120572 120573 120582 120575) and then
10038161003816100381610038161198861198991003816100381610038161003816 le 1 minus 120572 minus 119896 (1 minus 120573)Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 119899 ge 2 (40)
Set
120583119899 = Π119899 (1 + 119896) minus (119896120573 + 120572)Ω1198991 minus 120572 minus 119896 (1 minus 120573) 10038161003816100381610038161198861198991003816100381610038161003816 119899 ge 2 (41)
and 1205831 = 1minussuminfin119899=2 120583119899 119899 ge 2Then119891(119911) = suminfin119899=1 120583119899119891119899120578 and thiscompletes the proof
Theorem 8 Let 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Then 119891(119911) is closeto convex of order 120590(0 le 120590 lt 1) in the disc |119911| lt 1199031 where1199031 = inf [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)119899 (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(42)
Proof As 119891 isin 119881120578 where 119891 is close to convex of order 120590 wehave 100381610038161003816100381610038161198911015840 (119911) minus 110038161003816100381610038161003816 lt 1 minus 120590 (43)
as
100381610038161003816100381610038161198911015840 (119911) minus 110038161003816100381610038161003816 le infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 (44)
this expression is less than 1 minus 120590 ifinfinsum119899=2
1198991 minus 120590 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 lt 1 (45)
By the fact that 119891 isin 119896 minusVU120578(120572 120573 120582 120575) if and only if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 119886119899 le 1 (46)
inequality (43) is true if
1198991 minus 120590119911119899minus1 le Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) (47)
or equivalently
|119911|119899minus1 = [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)119899 (1 minus 120572 minus 119896 (1 minus 120573)) ] (48)
Theorem 9 Let 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Then 119891(119911) is closeto convex of order 120590 (0 le 120590 lt 1) in the disc|119911| lt 1199032 where1199032 = inf [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(49)
Proof As 119891 isin 119881120578 and 119891 is starlike of order 120590 then we have
100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1
100381610038161003816100381610038161003816100381610038161003816 lt 1 minus 120590 (50)
as 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1
100381610038161003816100381610038161003816100381610038161003816 lesuminfin119899=2 (119899 minus 1) 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus11 minus suminfin119899=2 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 (51)
The last expression is less than 1 minus 120590 ifinfinsum119899=2
119899 minus 1205901 minus 120590 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 lt 1 (52)
Using the fact that 119891 isin 119896 minusVU120578(120572 120573 120582 120575) if and only if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 119886119899 le 1 (53)
(50) is true if
119899 minus 1205901 minus 120590 |119911|119899minus1 lt Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) (54)
Or equivalently
|119911|119899minus1 = (1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) (55)
which is required
Theorem 10 Let119891 isin 119896minusVU120578(120572 120573 120582 120575)Then119891(119911) is convexof order 120590 (0 le 120590 lt 1) in the disc|119911| lt 1199033 where1199033 = inf [(1 minus 120590) [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]119899 (119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(56)
Proof Using the fact that 119891 is convex if and only if 1199111198911015840is starlike following the lines of Theorem 9 we have therequired results
Theorem 11 Let 119891119895(119911) (119895 = 1 2 ) given by (6) be in class119896 minusVU120578(120572 120573 120582 120575) Then (1198911 lowast 1198912) isin 119896 minusVU120578(1206011 120582 120575) for
6 Journal of Function Spaces
1206011 = (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus (Π119899 (1 + 119896) minus 119896120573Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))2(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus Ω119899 (1 minus 120572 minus 119896 (1 minus 120573))2 (57)
Proof We need to prove the largest 1206011 such that
(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(1 minus 1206011 minus 119896 (1 minus 120573)) 11988611989911198861198992 le 1 (58)
FromTheorem 3 we haveinfinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 1198861198991 le 1infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 1198861198992 le 1(59)
By Cauchy-Schwarz inequality we have
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ]radic11988611989911198861198992 le 1 (60)
Thus it is sufficient to show
[Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899(1 minus 1206011 minus 119896 (1 minus 120573)) ] 11988611989911198861198992le [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ]radic11988611989911198861198992 119899 ge 2
(61)
For 119899 ge 2radic11988611989911198861198992le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (1 minus 1206011 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))
(62)
Note that
radic11988611989911198861198992 le (1 minus 120572 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (63)
We need to show
(1 minus 120572 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (1 minus 1206011 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))
(64)
or equivalently
1206011 le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus (Π119899 (1 + 119896) minus 119896120573Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))2(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus Ω119899 (1 minus 120572 minus 119896 (1 minus 120573))2 = 120596 (119899) (65)
120596(119899) is an increasing function for 119899 ge 2 For 119899 = 2 in (65)
1206011 le 120596 (2) = (Π2 (1 + 119896) minus (119896120573 + 120572)Ω2)2 minus (Π2 (1 + 119896) minus 119896120573Ω2) (1 minus 120572 minus 119896 (1 minus 120573))2(Π2 (1 + 119896) minus (119896120573 + 120572)Ω2)2 minus Ω2 (1 minus 120572 minus 119896 (1 minus 120573))2 (66)
which proves main assertion of Theorem 11
Conflicts of Interest
The authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors jointly work on the results and they read andapproved the final manuscript
Acknowledgments
The work here is supported by MOHE Grant FRGS12016STG06UKM011
References
[1] A W Goodman ldquoOn uniformly convex functionsrdquo AnnalesPolonici Mathematici vol 56 no 1 pp 87ndash92 1991
[2] A W Goodman ldquoOn uniformly starlike functionsrdquo Journalof Mathematical Analysis and Applications vol 155 no 2 pp364ndash370 1991
Journal of Function Spaces 7
[3] W CMa andDMinda ldquoUniformly convex functionsrdquoAnnalesPolonici Mathematici vol 57 no 2 pp 165ndash175 1992
[4] F Roslashnning ldquoUniformly convex functions and a correspondingclass of starlike functionsrdquo Proceedings of the American Mathe-matical Society vol 118 no 1 pp 189ndash196 1993
[5] J Sokol and A Wisniowska-Wajnryb ldquoOn some classes ofstarlike functions related with parabolardquo Folia Sci Univ TechResov vol 121 no 18 pp 35ndash42 1993
[6] J Sokol and A Wisniowska-Wajnryb ldquoOn certain problem inthe classes of k-starlike functionsrdquo Computers amp Mathematicswith Applications vol 62 no 12 pp 4733ndash4741 2011
[7] S Kanas and A Wisniowska ldquoConic regions and k-uniformconvexityrdquo Journal of Computational and Applied Mathematicsvol 105 no 1-2 pp 327ndash336 1999
[8] S Kanas andHM Srivastava ldquoLinear operators associatedwithk-uniformly convex functionsrdquo Integral Transforms and SpecialFunction vol 9 no 2 pp 121ndash132 2000
[9] AMannino ldquoSome inequalities concerning starlike and convexfunctionsrdquo General Mathematics vol 12 no 1 pp 5ndash12 2004
[10] S Ponnusamy and M Vuorinen ldquoUnivalence and convexityproperties for Gaussian hypergeometric functionsrdquo The RockyMountain Journal of Mathematics vol 31 no 1 pp 327ndash3532001
[11] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975
[12] F Ronning ldquoIntegral representation for bounded starlike func-tionsrdquoAnnales Polonici Mathematici vol 60 no 3 pp 289ndash2971995
[13] R M El-Ashwah M K Aouf A A Hassan and A H HassanldquoCertain new classes of analytic functions with varying argu-mentsrdquo Journal of Complex Analysis vol 2013 Article ID958210 5 pages 2013
[14] RM Ali S R Mondal and V Ravichandran ldquoOn the Janowskiconvexity and starlikeness of the confluent hypergeometricfunctionrdquo Bulletin of the Belgian Mathematical Society SimonStevin vol 22 no 2 pp 227ndash250 2015
[15] R M Ali V Ravichandran and N Seenivasagan ldquoSubordina-tion and superordination of the Liu-Srivastava linear operatoron meromorphic functionsrdquo Bulletin of the Malaysian Mathe-matical Sciences Society vol 31 no 2 pp 193ndash207 2008
[16] R M Ali and V Ravichandran ldquoUniformly convex and uni-formly starlike functionsrdquo Mathematics Newsletter vol 21 pp16ndash30 2011
[17] S Altınkaya and S Yalcın ldquoCoefficient estimates for two newsubclasses of bi-univalent functions with respect to symmetricpointsrdquo Journal of Function Spaces Article ID 145242 2014
[18] M K Aouf H M Hossen and A Y Lashin ldquoOn certain fam-ilies of analytic functions with negative coefficientsrdquo IndianJournal of Pure and AppliedMathematics vol 31 no 8 pp 999ndash1015 2000
[19] M K Aouf A A Shamandy A O Mostafa and A K WagdyldquoCertain subclasses of uniformly starlike and convex functionsdefined by convolution with negative coefficientsrdquo Matem-atichki Vesnik vol 65 no 1 pp 14ndash28 2013
[20] A Kaminski and S Mincheva-Kaminska ldquoCompatibility con-ditions and the convolution of functions and generalized func-tionsrdquo Journal of Function Spaces and Applications vol 2013Article ID 356724 11 pages 2013
[21] N Magesh ldquoCertain subclasses of uniformly convex functionsof order 120572 and type 120573 with varying argumentsrdquo Journal of theEgyptian Mathematical Society vol 21 no 3 pp 184ndash189 2013
[22] K I Noor ldquoSome properties of certain analytic functionsrdquoJournal of Natural Geometry vol 7 no 1 pp 11ndash20 1995
[23] R Bharati R Parvatham and A Swaminathan ldquoOn subclassesof uniformly convex functions and corresponding class of star-like functionsrdquo Tamkang Journal of Mathematics vol 28 no 1pp 17ndash32 1997
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Function Spaces
The function
119891119899120578 (119911) = 119911 minus (1 minus 120572 minus 119896 (1 minus 120573)) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 1199111198990 le 120578 le 2120587 119899 ge 2
(26)
is extremal function
Corollary 4 Let 119891(119911) given in (1) be in class 119896 minusVU120578(120572 120573120582 120575) Then
10038161003816100381610038161198861198991003816100381610038161003816 le 1 minus 120572 minus 119896 (1 minus 120573)Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 119899 ge 2 (27)
Inequality (27) is attained for the function given in (26)
Theorem 5 Let the function 119891(119911) given in (1) be in class 119896 minusVU120578(120572 120573 120582 120575)Then for |119911| lt 119903 = 1
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge 119903 minus 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 11990321003816100381610038161003816119891 (119911)1003816100381610038161003816 le 119903 + 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 1199032
(28)
The results in (28) are attained for the function given in(26) for 119911 = plusmn119903Proof As we know fromTheorem 3
[Π2 (1 + 119896) minus (119896120573 + 120572)Ω2] infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816le infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(29)
As
1003816100381610038161003816119891 (119911)1003816100381610038161003816 ge |119911| minus infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899 ge 119903 minus 1199032 infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816ge 119903 minus 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 1199032
(30)
similarly
1003816100381610038161003816119891 (119911)1003816100381610038161003816 le |119911| + infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899 le 119903 + 1199032 infinsum119899=2
10038161003816100381610038161198861198991003816100381610038161003816le 119903 + 1 minus 120572 minus 119896 (1 minus 120573)Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 1199032
(31)
This completes the proof
Theorem 6 Let the function 119891(119911) given in (1) be in class 119896 minusVU120578(120572 120573 120582 120575)Then for |119911| lt 119903 = 1
1 minus 2 (1 minus 120572 minus 119896 (1 minus 120573))Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 119903 le 1003816100381610038161003816119891 (119911)1003816100381610038161003816le 1 + 2 (1 minus 120572 minus 119896 (1 minus 120573))Π2 (1 + 119896) minus (119896120573 + 120572)Ω2 119903
(32)
Proof For 119891(119911) given by (1) we have
100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 ge 1 minus infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 ge 1 minus 119903infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 le 1 + infinsum
119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 le 1 + 119903infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 (33)
In view of Theorem 3[Π2 (1 + 119896) minus (119896120573 + 120572)Ω2]2infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816le infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899] 10038161003816100381610038161198861198991003816100381610038161003816le 1 minus 120572 minus 119896 (1 minus 120573)
(34)
or equivalentlyinfinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 le 2 (1 minus 120572 minus 119896 (1 minus 120573))[Π2 (1 + 119896) minus (119896120573 + 120572)Ω2] (35)
A substitution from (35) into (33) yields inequality (32)which is required
Theorem 7 Let 119891 isin 119896 minus VU120578(120572 120573 120582 120575) with argumentproperty as in class 119881120578 Define 119891119895(119911) = 119911 and
119891119899120578 = 119911 minus 1 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]119911119899 (36)
where 0 le 120578 le 2120587 119899 ge 2Then function 119891(119911) is in class 119896 minus VU120578(120572 120573 120582 120575) if and
only if it can be expressed as
119891 (119911) = infinsum119899=1
120583119899119891119899120578 (37)
where 120583119899 ge 0 (119899 ge 1) and suminfin119899=1 120583119899 = 1Proof Assume that
119891 (119911) = 12058311198911 (119911)+ infinsum119899=2
120583119899 [119911 minus 1 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 119911119899]= infinsum119899=1
120583119899119911minus infinsum119899=2
[ 1 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]120583119899119911119899
(38)
Journal of Function Spaces 5
Then it follows thatinfinsum119899=2
10038161003816100381610038161003816100381610038161003816100381610038161 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899
1003816100381610038161003816100381610038161003816100381610038161003816sdot 120583119899 [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]= infinsum119899=2
120583119899 [1 minus 120572 minus 119896 (1 minus 120573)] le (1 minus 1205831)sdot [1 minus 120572 minus 119896 (1 minus 120573)] le 1 minus 120572 minus 119896 (1 minus 120573)
(39)
by Theorem 3 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Conversely assumethat the function 119891(119911) defined by (1) belongs to class 119896 minusVU120578(120572 120573 120582 120575) and then
10038161003816100381610038161198861198991003816100381610038161003816 le 1 minus 120572 minus 119896 (1 minus 120573)Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 119899 ge 2 (40)
Set
120583119899 = Π119899 (1 + 119896) minus (119896120573 + 120572)Ω1198991 minus 120572 minus 119896 (1 minus 120573) 10038161003816100381610038161198861198991003816100381610038161003816 119899 ge 2 (41)
and 1205831 = 1minussuminfin119899=2 120583119899 119899 ge 2Then119891(119911) = suminfin119899=1 120583119899119891119899120578 and thiscompletes the proof
Theorem 8 Let 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Then 119891(119911) is closeto convex of order 120590(0 le 120590 lt 1) in the disc |119911| lt 1199031 where1199031 = inf [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)119899 (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(42)
Proof As 119891 isin 119881120578 where 119891 is close to convex of order 120590 wehave 100381610038161003816100381610038161198911015840 (119911) minus 110038161003816100381610038161003816 lt 1 minus 120590 (43)
as
100381610038161003816100381610038161198911015840 (119911) minus 110038161003816100381610038161003816 le infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 (44)
this expression is less than 1 minus 120590 ifinfinsum119899=2
1198991 minus 120590 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 lt 1 (45)
By the fact that 119891 isin 119896 minusVU120578(120572 120573 120582 120575) if and only if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 119886119899 le 1 (46)
inequality (43) is true if
1198991 minus 120590119911119899minus1 le Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) (47)
or equivalently
|119911|119899minus1 = [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)119899 (1 minus 120572 minus 119896 (1 minus 120573)) ] (48)
Theorem 9 Let 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Then 119891(119911) is closeto convex of order 120590 (0 le 120590 lt 1) in the disc|119911| lt 1199032 where1199032 = inf [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(49)
Proof As 119891 isin 119881120578 and 119891 is starlike of order 120590 then we have
100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1
100381610038161003816100381610038161003816100381610038161003816 lt 1 minus 120590 (50)
as 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1
100381610038161003816100381610038161003816100381610038161003816 lesuminfin119899=2 (119899 minus 1) 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus11 minus suminfin119899=2 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 (51)
The last expression is less than 1 minus 120590 ifinfinsum119899=2
119899 minus 1205901 minus 120590 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 lt 1 (52)
Using the fact that 119891 isin 119896 minusVU120578(120572 120573 120582 120575) if and only if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 119886119899 le 1 (53)
(50) is true if
119899 minus 1205901 minus 120590 |119911|119899minus1 lt Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) (54)
Or equivalently
|119911|119899minus1 = (1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) (55)
which is required
Theorem 10 Let119891 isin 119896minusVU120578(120572 120573 120582 120575)Then119891(119911) is convexof order 120590 (0 le 120590 lt 1) in the disc|119911| lt 1199033 where1199033 = inf [(1 minus 120590) [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]119899 (119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(56)
Proof Using the fact that 119891 is convex if and only if 1199111198911015840is starlike following the lines of Theorem 9 we have therequired results
Theorem 11 Let 119891119895(119911) (119895 = 1 2 ) given by (6) be in class119896 minusVU120578(120572 120573 120582 120575) Then (1198911 lowast 1198912) isin 119896 minusVU120578(1206011 120582 120575) for
6 Journal of Function Spaces
1206011 = (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus (Π119899 (1 + 119896) minus 119896120573Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))2(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus Ω119899 (1 minus 120572 minus 119896 (1 minus 120573))2 (57)
Proof We need to prove the largest 1206011 such that
(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(1 minus 1206011 minus 119896 (1 minus 120573)) 11988611989911198861198992 le 1 (58)
FromTheorem 3 we haveinfinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 1198861198991 le 1infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 1198861198992 le 1(59)
By Cauchy-Schwarz inequality we have
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ]radic11988611989911198861198992 le 1 (60)
Thus it is sufficient to show
[Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899(1 minus 1206011 minus 119896 (1 minus 120573)) ] 11988611989911198861198992le [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ]radic11988611989911198861198992 119899 ge 2
(61)
For 119899 ge 2radic11988611989911198861198992le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (1 minus 1206011 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))
(62)
Note that
radic11988611989911198861198992 le (1 minus 120572 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (63)
We need to show
(1 minus 120572 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (1 minus 1206011 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))
(64)
or equivalently
1206011 le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus (Π119899 (1 + 119896) minus 119896120573Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))2(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus Ω119899 (1 minus 120572 minus 119896 (1 minus 120573))2 = 120596 (119899) (65)
120596(119899) is an increasing function for 119899 ge 2 For 119899 = 2 in (65)
1206011 le 120596 (2) = (Π2 (1 + 119896) minus (119896120573 + 120572)Ω2)2 minus (Π2 (1 + 119896) minus 119896120573Ω2) (1 minus 120572 minus 119896 (1 minus 120573))2(Π2 (1 + 119896) minus (119896120573 + 120572)Ω2)2 minus Ω2 (1 minus 120572 minus 119896 (1 minus 120573))2 (66)
which proves main assertion of Theorem 11
Conflicts of Interest
The authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors jointly work on the results and they read andapproved the final manuscript
Acknowledgments
The work here is supported by MOHE Grant FRGS12016STG06UKM011
References
[1] A W Goodman ldquoOn uniformly convex functionsrdquo AnnalesPolonici Mathematici vol 56 no 1 pp 87ndash92 1991
[2] A W Goodman ldquoOn uniformly starlike functionsrdquo Journalof Mathematical Analysis and Applications vol 155 no 2 pp364ndash370 1991
Journal of Function Spaces 7
[3] W CMa andDMinda ldquoUniformly convex functionsrdquoAnnalesPolonici Mathematici vol 57 no 2 pp 165ndash175 1992
[4] F Roslashnning ldquoUniformly convex functions and a correspondingclass of starlike functionsrdquo Proceedings of the American Mathe-matical Society vol 118 no 1 pp 189ndash196 1993
[5] J Sokol and A Wisniowska-Wajnryb ldquoOn some classes ofstarlike functions related with parabolardquo Folia Sci Univ TechResov vol 121 no 18 pp 35ndash42 1993
[6] J Sokol and A Wisniowska-Wajnryb ldquoOn certain problem inthe classes of k-starlike functionsrdquo Computers amp Mathematicswith Applications vol 62 no 12 pp 4733ndash4741 2011
[7] S Kanas and A Wisniowska ldquoConic regions and k-uniformconvexityrdquo Journal of Computational and Applied Mathematicsvol 105 no 1-2 pp 327ndash336 1999
[8] S Kanas andHM Srivastava ldquoLinear operators associatedwithk-uniformly convex functionsrdquo Integral Transforms and SpecialFunction vol 9 no 2 pp 121ndash132 2000
[9] AMannino ldquoSome inequalities concerning starlike and convexfunctionsrdquo General Mathematics vol 12 no 1 pp 5ndash12 2004
[10] S Ponnusamy and M Vuorinen ldquoUnivalence and convexityproperties for Gaussian hypergeometric functionsrdquo The RockyMountain Journal of Mathematics vol 31 no 1 pp 327ndash3532001
[11] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975
[12] F Ronning ldquoIntegral representation for bounded starlike func-tionsrdquoAnnales Polonici Mathematici vol 60 no 3 pp 289ndash2971995
[13] R M El-Ashwah M K Aouf A A Hassan and A H HassanldquoCertain new classes of analytic functions with varying argu-mentsrdquo Journal of Complex Analysis vol 2013 Article ID958210 5 pages 2013
[14] RM Ali S R Mondal and V Ravichandran ldquoOn the Janowskiconvexity and starlikeness of the confluent hypergeometricfunctionrdquo Bulletin of the Belgian Mathematical Society SimonStevin vol 22 no 2 pp 227ndash250 2015
[15] R M Ali V Ravichandran and N Seenivasagan ldquoSubordina-tion and superordination of the Liu-Srivastava linear operatoron meromorphic functionsrdquo Bulletin of the Malaysian Mathe-matical Sciences Society vol 31 no 2 pp 193ndash207 2008
[16] R M Ali and V Ravichandran ldquoUniformly convex and uni-formly starlike functionsrdquo Mathematics Newsletter vol 21 pp16ndash30 2011
[17] S Altınkaya and S Yalcın ldquoCoefficient estimates for two newsubclasses of bi-univalent functions with respect to symmetricpointsrdquo Journal of Function Spaces Article ID 145242 2014
[18] M K Aouf H M Hossen and A Y Lashin ldquoOn certain fam-ilies of analytic functions with negative coefficientsrdquo IndianJournal of Pure and AppliedMathematics vol 31 no 8 pp 999ndash1015 2000
[19] M K Aouf A A Shamandy A O Mostafa and A K WagdyldquoCertain subclasses of uniformly starlike and convex functionsdefined by convolution with negative coefficientsrdquo Matem-atichki Vesnik vol 65 no 1 pp 14ndash28 2013
[20] A Kaminski and S Mincheva-Kaminska ldquoCompatibility con-ditions and the convolution of functions and generalized func-tionsrdquo Journal of Function Spaces and Applications vol 2013Article ID 356724 11 pages 2013
[21] N Magesh ldquoCertain subclasses of uniformly convex functionsof order 120572 and type 120573 with varying argumentsrdquo Journal of theEgyptian Mathematical Society vol 21 no 3 pp 184ndash189 2013
[22] K I Noor ldquoSome properties of certain analytic functionsrdquoJournal of Natural Geometry vol 7 no 1 pp 11ndash20 1995
[23] R Bharati R Parvatham and A Swaminathan ldquoOn subclassesof uniformly convex functions and corresponding class of star-like functionsrdquo Tamkang Journal of Mathematics vol 28 no 1pp 17ndash32 1997
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 5
Then it follows thatinfinsum119899=2
10038161003816100381610038161003816100381610038161003816100381610038161 minus 120572 minus 119896 (1 minus 120573) 119890119894(1minus119899)120578Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899
1003816100381610038161003816100381610038161003816100381610038161003816sdot 120583119899 [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]= infinsum119899=2
120583119899 [1 minus 120572 minus 119896 (1 minus 120573)] le (1 minus 1205831)sdot [1 minus 120572 minus 119896 (1 minus 120573)] le 1 minus 120572 minus 119896 (1 minus 120573)
(39)
by Theorem 3 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Conversely assumethat the function 119891(119911) defined by (1) belongs to class 119896 minusVU120578(120572 120573 120582 120575) and then
10038161003816100381610038161198861198991003816100381610038161003816 le 1 minus 120572 minus 119896 (1 minus 120573)Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899 119899 ge 2 (40)
Set
120583119899 = Π119899 (1 + 119896) minus (119896120573 + 120572)Ω1198991 minus 120572 minus 119896 (1 minus 120573) 10038161003816100381610038161198861198991003816100381610038161003816 119899 ge 2 (41)
and 1205831 = 1minussuminfin119899=2 120583119899 119899 ge 2Then119891(119911) = suminfin119899=1 120583119899119891119899120578 and thiscompletes the proof
Theorem 8 Let 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Then 119891(119911) is closeto convex of order 120590(0 le 120590 lt 1) in the disc |119911| lt 1199031 where1199031 = inf [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)119899 (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(42)
Proof As 119891 isin 119881120578 where 119891 is close to convex of order 120590 wehave 100381610038161003816100381610038161198911015840 (119911) minus 110038161003816100381610038161003816 lt 1 minus 120590 (43)
as
100381610038161003816100381610038161198911015840 (119911) minus 110038161003816100381610038161003816 le infinsum119899=2
119899 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 (44)
this expression is less than 1 minus 120590 ifinfinsum119899=2
1198991 minus 120590 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 lt 1 (45)
By the fact that 119891 isin 119896 minusVU120578(120572 120573 120582 120575) if and only if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 119886119899 le 1 (46)
inequality (43) is true if
1198991 minus 120590119911119899minus1 le Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) (47)
or equivalently
|119911|119899minus1 = [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)119899 (1 minus 120572 minus 119896 (1 minus 120573)) ] (48)
Theorem 9 Let 119891 isin 119896 minusVU120578(120572 120573 120582 120575) Then 119891(119911) is closeto convex of order 120590 (0 le 120590 lt 1) in the disc|119911| lt 1199032 where1199032 = inf [(1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(49)
Proof As 119891 isin 119881120578 and 119891 is starlike of order 120590 then we have
100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1
100381610038161003816100381610038161003816100381610038161003816 lt 1 minus 120590 (50)
as 100381610038161003816100381610038161003816100381610038161003816 1199111198911015840 (119911)119891 (119911) minus 1
100381610038161003816100381610038161003816100381610038161003816 lesuminfin119899=2 (119899 minus 1) 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus11 minus suminfin119899=2 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 (51)
The last expression is less than 1 minus 120590 ifinfinsum119899=2
119899 minus 1205901 minus 120590 10038161003816100381610038161198861198991003816100381610038161003816 |119911|119899minus1 lt 1 (52)
Using the fact that 119891 isin 119896 minusVU120578(120572 120573 120582 120575) if and only if
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 119886119899 le 1 (53)
(50) is true if
119899 minus 1205901 minus 120590 |119911|119899minus1 lt Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) (54)
Or equivalently
|119911|119899minus1 = (1 minus 120590) (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) (55)
which is required
Theorem 10 Let119891 isin 119896minusVU120578(120572 120573 120582 120575)Then119891(119911) is convexof order 120590 (0 le 120590 lt 1) in the disc|119911| lt 1199033 where1199033 = inf [(1 minus 120590) [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899]119899 (119899 minus 120590) (1 minus 120572 minus 119896 (1 minus 120573)) ]1(119899minus1)
119899 ge 2(56)
Proof Using the fact that 119891 is convex if and only if 1199111198911015840is starlike following the lines of Theorem 9 we have therequired results
Theorem 11 Let 119891119895(119911) (119895 = 1 2 ) given by (6) be in class119896 minusVU120578(120572 120573 120582 120575) Then (1198911 lowast 1198912) isin 119896 minusVU120578(1206011 120582 120575) for
6 Journal of Function Spaces
1206011 = (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus (Π119899 (1 + 119896) minus 119896120573Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))2(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus Ω119899 (1 minus 120572 minus 119896 (1 minus 120573))2 (57)
Proof We need to prove the largest 1206011 such that
(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(1 minus 1206011 minus 119896 (1 minus 120573)) 11988611989911198861198992 le 1 (58)
FromTheorem 3 we haveinfinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 1198861198991 le 1infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 1198861198992 le 1(59)
By Cauchy-Schwarz inequality we have
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ]radic11988611989911198861198992 le 1 (60)
Thus it is sufficient to show
[Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899(1 minus 1206011 minus 119896 (1 minus 120573)) ] 11988611989911198861198992le [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ]radic11988611989911198861198992 119899 ge 2
(61)
For 119899 ge 2radic11988611989911198861198992le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (1 minus 1206011 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))
(62)
Note that
radic11988611989911198861198992 le (1 minus 120572 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (63)
We need to show
(1 minus 120572 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (1 minus 1206011 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))
(64)
or equivalently
1206011 le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus (Π119899 (1 + 119896) minus 119896120573Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))2(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus Ω119899 (1 minus 120572 minus 119896 (1 minus 120573))2 = 120596 (119899) (65)
120596(119899) is an increasing function for 119899 ge 2 For 119899 = 2 in (65)
1206011 le 120596 (2) = (Π2 (1 + 119896) minus (119896120573 + 120572)Ω2)2 minus (Π2 (1 + 119896) minus 119896120573Ω2) (1 minus 120572 minus 119896 (1 minus 120573))2(Π2 (1 + 119896) minus (119896120573 + 120572)Ω2)2 minus Ω2 (1 minus 120572 minus 119896 (1 minus 120573))2 (66)
which proves main assertion of Theorem 11
Conflicts of Interest
The authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors jointly work on the results and they read andapproved the final manuscript
Acknowledgments
The work here is supported by MOHE Grant FRGS12016STG06UKM011
References
[1] A W Goodman ldquoOn uniformly convex functionsrdquo AnnalesPolonici Mathematici vol 56 no 1 pp 87ndash92 1991
[2] A W Goodman ldquoOn uniformly starlike functionsrdquo Journalof Mathematical Analysis and Applications vol 155 no 2 pp364ndash370 1991
Journal of Function Spaces 7
[3] W CMa andDMinda ldquoUniformly convex functionsrdquoAnnalesPolonici Mathematici vol 57 no 2 pp 165ndash175 1992
[4] F Roslashnning ldquoUniformly convex functions and a correspondingclass of starlike functionsrdquo Proceedings of the American Mathe-matical Society vol 118 no 1 pp 189ndash196 1993
[5] J Sokol and A Wisniowska-Wajnryb ldquoOn some classes ofstarlike functions related with parabolardquo Folia Sci Univ TechResov vol 121 no 18 pp 35ndash42 1993
[6] J Sokol and A Wisniowska-Wajnryb ldquoOn certain problem inthe classes of k-starlike functionsrdquo Computers amp Mathematicswith Applications vol 62 no 12 pp 4733ndash4741 2011
[7] S Kanas and A Wisniowska ldquoConic regions and k-uniformconvexityrdquo Journal of Computational and Applied Mathematicsvol 105 no 1-2 pp 327ndash336 1999
[8] S Kanas andHM Srivastava ldquoLinear operators associatedwithk-uniformly convex functionsrdquo Integral Transforms and SpecialFunction vol 9 no 2 pp 121ndash132 2000
[9] AMannino ldquoSome inequalities concerning starlike and convexfunctionsrdquo General Mathematics vol 12 no 1 pp 5ndash12 2004
[10] S Ponnusamy and M Vuorinen ldquoUnivalence and convexityproperties for Gaussian hypergeometric functionsrdquo The RockyMountain Journal of Mathematics vol 31 no 1 pp 327ndash3532001
[11] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975
[12] F Ronning ldquoIntegral representation for bounded starlike func-tionsrdquoAnnales Polonici Mathematici vol 60 no 3 pp 289ndash2971995
[13] R M El-Ashwah M K Aouf A A Hassan and A H HassanldquoCertain new classes of analytic functions with varying argu-mentsrdquo Journal of Complex Analysis vol 2013 Article ID958210 5 pages 2013
[14] RM Ali S R Mondal and V Ravichandran ldquoOn the Janowskiconvexity and starlikeness of the confluent hypergeometricfunctionrdquo Bulletin of the Belgian Mathematical Society SimonStevin vol 22 no 2 pp 227ndash250 2015
[15] R M Ali V Ravichandran and N Seenivasagan ldquoSubordina-tion and superordination of the Liu-Srivastava linear operatoron meromorphic functionsrdquo Bulletin of the Malaysian Mathe-matical Sciences Society vol 31 no 2 pp 193ndash207 2008
[16] R M Ali and V Ravichandran ldquoUniformly convex and uni-formly starlike functionsrdquo Mathematics Newsletter vol 21 pp16ndash30 2011
[17] S Altınkaya and S Yalcın ldquoCoefficient estimates for two newsubclasses of bi-univalent functions with respect to symmetricpointsrdquo Journal of Function Spaces Article ID 145242 2014
[18] M K Aouf H M Hossen and A Y Lashin ldquoOn certain fam-ilies of analytic functions with negative coefficientsrdquo IndianJournal of Pure and AppliedMathematics vol 31 no 8 pp 999ndash1015 2000
[19] M K Aouf A A Shamandy A O Mostafa and A K WagdyldquoCertain subclasses of uniformly starlike and convex functionsdefined by convolution with negative coefficientsrdquo Matem-atichki Vesnik vol 65 no 1 pp 14ndash28 2013
[20] A Kaminski and S Mincheva-Kaminska ldquoCompatibility con-ditions and the convolution of functions and generalized func-tionsrdquo Journal of Function Spaces and Applications vol 2013Article ID 356724 11 pages 2013
[21] N Magesh ldquoCertain subclasses of uniformly convex functionsof order 120572 and type 120573 with varying argumentsrdquo Journal of theEgyptian Mathematical Society vol 21 no 3 pp 184ndash189 2013
[22] K I Noor ldquoSome properties of certain analytic functionsrdquoJournal of Natural Geometry vol 7 no 1 pp 11ndash20 1995
[23] R Bharati R Parvatham and A Swaminathan ldquoOn subclassesof uniformly convex functions and corresponding class of star-like functionsrdquo Tamkang Journal of Mathematics vol 28 no 1pp 17ndash32 1997
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Function Spaces
1206011 = (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus (Π119899 (1 + 119896) minus 119896120573Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))2(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus Ω119899 (1 minus 120572 minus 119896 (1 minus 120573))2 (57)
Proof We need to prove the largest 1206011 such that
(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)(1 minus 1206011 minus 119896 (1 minus 120573)) 11988611989911198861198992 le 1 (58)
FromTheorem 3 we haveinfinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 1198861198991 le 1infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ] 1198861198992 le 1(59)
By Cauchy-Schwarz inequality we have
infinsum119899=2
[Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ]radic11988611989911198861198992 le 1 (60)
Thus it is sufficient to show
[Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899(1 minus 1206011 minus 119896 (1 minus 120573)) ] 11988611989911198861198992le [Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899(1 minus 120572 minus 119896 (1 minus 120573)) ]radic11988611989911198861198992 119899 ge 2
(61)
For 119899 ge 2radic11988611989911198861198992le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (1 minus 1206011 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))
(62)
Note that
radic11988611989911198861198992 le (1 minus 120572 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (63)
We need to show
(1 minus 120572 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899) (1 minus 1206011 minus 119896 (1 minus 120573))(Π119899 (1 + 119896) minus (119896120573 + 1206011)Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))
(64)
or equivalently
1206011 le (Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus (Π119899 (1 + 119896) minus 119896120573Ω119899) (1 minus 120572 minus 119896 (1 minus 120573))2(Π119899 (1 + 119896) minus (119896120573 + 120572)Ω119899)2 minus Ω119899 (1 minus 120572 minus 119896 (1 minus 120573))2 = 120596 (119899) (65)
120596(119899) is an increasing function for 119899 ge 2 For 119899 = 2 in (65)
1206011 le 120596 (2) = (Π2 (1 + 119896) minus (119896120573 + 120572)Ω2)2 minus (Π2 (1 + 119896) minus 119896120573Ω2) (1 minus 120572 minus 119896 (1 minus 120573))2(Π2 (1 + 119896) minus (119896120573 + 120572)Ω2)2 minus Ω2 (1 minus 120572 minus 119896 (1 minus 120573))2 (66)
which proves main assertion of Theorem 11
Conflicts of Interest
The authors declare that they have no conflicts of interest
Authorsrsquo Contributions
All authors jointly work on the results and they read andapproved the final manuscript
Acknowledgments
The work here is supported by MOHE Grant FRGS12016STG06UKM011
References
[1] A W Goodman ldquoOn uniformly convex functionsrdquo AnnalesPolonici Mathematici vol 56 no 1 pp 87ndash92 1991
[2] A W Goodman ldquoOn uniformly starlike functionsrdquo Journalof Mathematical Analysis and Applications vol 155 no 2 pp364ndash370 1991
Journal of Function Spaces 7
[3] W CMa andDMinda ldquoUniformly convex functionsrdquoAnnalesPolonici Mathematici vol 57 no 2 pp 165ndash175 1992
[4] F Roslashnning ldquoUniformly convex functions and a correspondingclass of starlike functionsrdquo Proceedings of the American Mathe-matical Society vol 118 no 1 pp 189ndash196 1993
[5] J Sokol and A Wisniowska-Wajnryb ldquoOn some classes ofstarlike functions related with parabolardquo Folia Sci Univ TechResov vol 121 no 18 pp 35ndash42 1993
[6] J Sokol and A Wisniowska-Wajnryb ldquoOn certain problem inthe classes of k-starlike functionsrdquo Computers amp Mathematicswith Applications vol 62 no 12 pp 4733ndash4741 2011
[7] S Kanas and A Wisniowska ldquoConic regions and k-uniformconvexityrdquo Journal of Computational and Applied Mathematicsvol 105 no 1-2 pp 327ndash336 1999
[8] S Kanas andHM Srivastava ldquoLinear operators associatedwithk-uniformly convex functionsrdquo Integral Transforms and SpecialFunction vol 9 no 2 pp 121ndash132 2000
[9] AMannino ldquoSome inequalities concerning starlike and convexfunctionsrdquo General Mathematics vol 12 no 1 pp 5ndash12 2004
[10] S Ponnusamy and M Vuorinen ldquoUnivalence and convexityproperties for Gaussian hypergeometric functionsrdquo The RockyMountain Journal of Mathematics vol 31 no 1 pp 327ndash3532001
[11] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975
[12] F Ronning ldquoIntegral representation for bounded starlike func-tionsrdquoAnnales Polonici Mathematici vol 60 no 3 pp 289ndash2971995
[13] R M El-Ashwah M K Aouf A A Hassan and A H HassanldquoCertain new classes of analytic functions with varying argu-mentsrdquo Journal of Complex Analysis vol 2013 Article ID958210 5 pages 2013
[14] RM Ali S R Mondal and V Ravichandran ldquoOn the Janowskiconvexity and starlikeness of the confluent hypergeometricfunctionrdquo Bulletin of the Belgian Mathematical Society SimonStevin vol 22 no 2 pp 227ndash250 2015
[15] R M Ali V Ravichandran and N Seenivasagan ldquoSubordina-tion and superordination of the Liu-Srivastava linear operatoron meromorphic functionsrdquo Bulletin of the Malaysian Mathe-matical Sciences Society vol 31 no 2 pp 193ndash207 2008
[16] R M Ali and V Ravichandran ldquoUniformly convex and uni-formly starlike functionsrdquo Mathematics Newsletter vol 21 pp16ndash30 2011
[17] S Altınkaya and S Yalcın ldquoCoefficient estimates for two newsubclasses of bi-univalent functions with respect to symmetricpointsrdquo Journal of Function Spaces Article ID 145242 2014
[18] M K Aouf H M Hossen and A Y Lashin ldquoOn certain fam-ilies of analytic functions with negative coefficientsrdquo IndianJournal of Pure and AppliedMathematics vol 31 no 8 pp 999ndash1015 2000
[19] M K Aouf A A Shamandy A O Mostafa and A K WagdyldquoCertain subclasses of uniformly starlike and convex functionsdefined by convolution with negative coefficientsrdquo Matem-atichki Vesnik vol 65 no 1 pp 14ndash28 2013
[20] A Kaminski and S Mincheva-Kaminska ldquoCompatibility con-ditions and the convolution of functions and generalized func-tionsrdquo Journal of Function Spaces and Applications vol 2013Article ID 356724 11 pages 2013
[21] N Magesh ldquoCertain subclasses of uniformly convex functionsof order 120572 and type 120573 with varying argumentsrdquo Journal of theEgyptian Mathematical Society vol 21 no 3 pp 184ndash189 2013
[22] K I Noor ldquoSome properties of certain analytic functionsrdquoJournal of Natural Geometry vol 7 no 1 pp 11ndash20 1995
[23] R Bharati R Parvatham and A Swaminathan ldquoOn subclassesof uniformly convex functions and corresponding class of star-like functionsrdquo Tamkang Journal of Mathematics vol 28 no 1pp 17ndash32 1997
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 7
[3] W CMa andDMinda ldquoUniformly convex functionsrdquoAnnalesPolonici Mathematici vol 57 no 2 pp 165ndash175 1992
[4] F Roslashnning ldquoUniformly convex functions and a correspondingclass of starlike functionsrdquo Proceedings of the American Mathe-matical Society vol 118 no 1 pp 189ndash196 1993
[5] J Sokol and A Wisniowska-Wajnryb ldquoOn some classes ofstarlike functions related with parabolardquo Folia Sci Univ TechResov vol 121 no 18 pp 35ndash42 1993
[6] J Sokol and A Wisniowska-Wajnryb ldquoOn certain problem inthe classes of k-starlike functionsrdquo Computers amp Mathematicswith Applications vol 62 no 12 pp 4733ndash4741 2011
[7] S Kanas and A Wisniowska ldquoConic regions and k-uniformconvexityrdquo Journal of Computational and Applied Mathematicsvol 105 no 1-2 pp 327ndash336 1999
[8] S Kanas andHM Srivastava ldquoLinear operators associatedwithk-uniformly convex functionsrdquo Integral Transforms and SpecialFunction vol 9 no 2 pp 121ndash132 2000
[9] AMannino ldquoSome inequalities concerning starlike and convexfunctionsrdquo General Mathematics vol 12 no 1 pp 5ndash12 2004
[10] S Ponnusamy and M Vuorinen ldquoUnivalence and convexityproperties for Gaussian hypergeometric functionsrdquo The RockyMountain Journal of Mathematics vol 31 no 1 pp 327ndash3532001
[11] H Silverman ldquoUnivalent functions with negative coefficientsrdquoProceedings of the American Mathematical Society vol 51 pp109ndash116 1975
[12] F Ronning ldquoIntegral representation for bounded starlike func-tionsrdquoAnnales Polonici Mathematici vol 60 no 3 pp 289ndash2971995
[13] R M El-Ashwah M K Aouf A A Hassan and A H HassanldquoCertain new classes of analytic functions with varying argu-mentsrdquo Journal of Complex Analysis vol 2013 Article ID958210 5 pages 2013
[14] RM Ali S R Mondal and V Ravichandran ldquoOn the Janowskiconvexity and starlikeness of the confluent hypergeometricfunctionrdquo Bulletin of the Belgian Mathematical Society SimonStevin vol 22 no 2 pp 227ndash250 2015
[15] R M Ali V Ravichandran and N Seenivasagan ldquoSubordina-tion and superordination of the Liu-Srivastava linear operatoron meromorphic functionsrdquo Bulletin of the Malaysian Mathe-matical Sciences Society vol 31 no 2 pp 193ndash207 2008
[16] R M Ali and V Ravichandran ldquoUniformly convex and uni-formly starlike functionsrdquo Mathematics Newsletter vol 21 pp16ndash30 2011
[17] S Altınkaya and S Yalcın ldquoCoefficient estimates for two newsubclasses of bi-univalent functions with respect to symmetricpointsrdquo Journal of Function Spaces Article ID 145242 2014
[18] M K Aouf H M Hossen and A Y Lashin ldquoOn certain fam-ilies of analytic functions with negative coefficientsrdquo IndianJournal of Pure and AppliedMathematics vol 31 no 8 pp 999ndash1015 2000
[19] M K Aouf A A Shamandy A O Mostafa and A K WagdyldquoCertain subclasses of uniformly starlike and convex functionsdefined by convolution with negative coefficientsrdquo Matem-atichki Vesnik vol 65 no 1 pp 14ndash28 2013
[20] A Kaminski and S Mincheva-Kaminska ldquoCompatibility con-ditions and the convolution of functions and generalized func-tionsrdquo Journal of Function Spaces and Applications vol 2013Article ID 356724 11 pages 2013
[21] N Magesh ldquoCertain subclasses of uniformly convex functionsof order 120572 and type 120573 with varying argumentsrdquo Journal of theEgyptian Mathematical Society vol 21 no 3 pp 184ndash189 2013
[22] K I Noor ldquoSome properties of certain analytic functionsrdquoJournal of Natural Geometry vol 7 no 1 pp 11ndash20 1995
[23] R Bharati R Parvatham and A Swaminathan ldquoOn subclassesof uniformly convex functions and corresponding class of star-like functionsrdquo Tamkang Journal of Mathematics vol 28 no 1pp 17ndash32 1997
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
