Bessel Function with Linear Differential Operator · 2021. 1. 22. · Bessel Function with Linear...

Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 16, Number 5 (2020), pp. 623-639 © Research India Publications http://www.ripublication.com/gjpam.htm

Bessel Function with Linear Differential Operator

S. Lakshmi1*, K. R. Karthikeyan2, S. Varadharajan3 and C. Selvaraj4

2 National University of Science and Technology, Muscat, Sultanate of Oman.

3 University of Technology and Applied Science – Al Musanna, Sultanate of Oman.

4 Presidency College (Autonoumous), Chennai, India.

Abstract

In this paper, a class of analytic functions f defined on the open unit disc satisfying

2

1)(

)(1>

2

1)(

)(11

zf

z'zf

zf

z'zfieRe

is studied, among other results, inclusion relations and applications involving a certain class of integral operator are also considered.

Keywords: Analytic function, univalent function, starlike function, convex function, subordination, bessel function

2010 Mathematics Subject Classification: 30C45, 30C50.

1 INTRODUCTION

Let A denote the class of all analytic function of the form

n

n

n

zazzf

2=

=)( (1.1)

in the open unit disc 1|

624 S. Lakshmi, K. R. Karthikeyan, S. Varadharajan and C. Selvaraj

of univalent functions in U . We say that the function f is convex when )(Uf is a convex set. Also, we say that a function f is starlike with respect to the origin when

)(Uf is a starlike set with respect to 0. By K or *S we denote the subclasses of A

consisting of all functions which are convex or starlike respectively, while by )(S* we denote the class of starlike functions of order 0,1, .

In 1991, Goodman [7] introduced the class UCV of uniformly convex functions. A function CVf is the class UCV if for every circular are U , with center in U , the arc )(f is convex. A more useful characterization of class UCV was given by Ma and Minda [11] (see also [17]) as:

).(,)(

)(>

)(

)(1 Uz

z'f

z''zf

z'f

z''zfReandff

AUCV

In 1999, Kanas and Wisniowska [8] (see also [9]) introduced the class of k -uniformaly convex functions, 0k , denoted by k - UCV and the class k - ST related to k -UCV by Alexandar type relation, i.e.,

).(,)(

)(>

)(

)(1 Uz

z'f

z''zfk

z'f

z''zfReandfk'zfkf

ASTUCV

In [8] and [9] respectively, their geometric definitions and connections with the conic domains were also considered. For a fixed 0k , the class k -UCV is defined purely geometrically as a subclass of univalent functions which map the intersection of U with any disk centered at k||, , onto a convex domain. The notion of k -uniform convexity is a natural extension of the classical convexity. Observe that, if 0=k then the center is the origin and the class k - UCV reduces to the class Vk . Moreover for 1=k it coincides with the class of uniformly convex functions UCV introduced by Goodman [7] and studied extensively by R ø nning [17] and independently by Ma and Minda [11]. The class k - UCV started much earlier in papers [5, 6] with some additional conditions but without the geometric interpretation.

We say that a function Af is in the {0}\0,,*, CS kk , if and only if

).(,1)(

)(1>1

)(

)(11 Uz

zf

z'zfk

zf

z'zfRe

Bessel Function with Linear Differential Operator 625

A lot of authors investigated the properties of the class * ,kS and their generalizations

in several directions, e.g., see [1, 2, 6, 9, 15, 17, 19]. An analytic function f is said to be subordinate to an analytic function g (written as gf ) if and only if there exists an analytic function with

Uzforzand 1|


defined by

).(21!

1)(=)(

2

0=

CJ

zz

nvnz

vnn

n

v

(1.4)

ii) For 1=b and 1= d in (1.3), we obtain the modified Bessel function of the first kind of order v defined by

).(21!

1)(=)(

2

0=

CI

zz

nvnz

vnn

n

v

(1.5)

iii) For 2=b and 1=d in (1.3), the function )(,, zdbv reduces to )(2

zvS

,

where vS is the spherical Bessel function of the first kind of order v , defined by

).(2

2

3!

1)(

2=)(

2

0=

CS

zz

nvn

z

vnn

n

v

(1.6)

Now, consider the function CC:)(,, zu dbv , defined by the transformation

).(2

12=)( ,,

2,, zz

bvzu dbv

v

v

dbv

(1.7)

By using the well-known Pochhammer symbol (or the shifted factorial) )( defined,

for C, and in terms of the Euler function, by

),;=(1)(1)(

{0}),\0;=(1=

)(

)(:=)(

CN

C

nn

and 1=)( 0 , we obtain for the function )(,, zu dbv the following representation

,!

2

1

4=)(

0

,,n

z

bv

d

zun

n

n

n

dbv

where 2,1,0,2

1=

bvk . This function is analytic on C and satisfies the


second order linear differential equation

0.=)()(1)2(2)(24 zdzuz'zubvz''uz

Now, we introduce the function )(,, zdbv defined in terms of generalized Bessel

function )(,, zdbv , defined by

)(=)( ,,,, zzuz dbvdbv

)(2

12= ,,

21

zzb

v dbv

v

v

2

1=,

)(!4

)(=

1

1=

bvkwhere

kn

zdz

n

n

nn

n

),,(= zdkg

Motivated by Selvaraj and Karthikeyan [14], we define the following UUzfdkDm :)(),( by

),,()(=)(),( zdkgzfzfdkD (1.8)

'zdkgzfzzdkgzfzfdkD )),,()((),,()()(1=)(),(1 (1.9)

)(),(=)(),( 11 zfdkDDzfdkD mm (1.10)

If Af , then from (1.9) and (1.10) we may easily deduce that

1

1

1

2= )(1)!(4

)(1)(1=)(),(

n

n

n

n

nm

n

m

kn

zadnzzfdkD

(1.11)

where 0=0 NNm and 0 .

It can be easily verified from definition of (1.11) that

)(),()(1)(),(1=)(),( zfdkmDzfdkmD'

zfdkmDz

(1.12)


In the special cases of the )(),( zfdkDm , we obtain the following operators related to the Bessel function:

i) Choosing 0=m in (1.11) we get the Deniz operator

1

1

1

2= )(1)!(4

)(==

n

n

n

n

n

n

c

kkn

zadzB .

ii) Choosing 0=m and 1== db in (1.11) we obtain the operator AAJ :v related with Bessel function, defined by

.1)(1)!(4

1)(=)(

1

1

1

2=

n

n

n

n

n

n

vvn

zazzfJ

iii) Choosing 0=m , 1=b and 1= d in (1.11) we obtain the operator AAJ :v related with modified Bessel function, defined by

.1)(1)!(4

(1)=)(

1

1

1

2=

n

n

n

n

n

n

vvn

zazzfJ

iv) Choosing 0=m , 2=b and 1=d in (1.11) we obtain the operator AAS :v related with spherical Bessel function, defined by

.

2

31)!(4

1)(=)(

1

1

1

2=

n

n

n

n

n

n

v

vn

zazzfS

Definition 1.1 A function A)(zf is in the class ,,S , {0}\C if and only if

2

1)(

)(1>

2

1)(

)(11

zf

z'zf

zf

z'zfieRe

. (1.13)


Remark 1 The above differential inequality can be equivalently written as

)(=sincos)(1)(

)(11 zKizhie

zf

z'zf

,

where

2

2

)(1

)(1

)(1

)(1

2

cos1=)(

z

z

z

z

logzh

, 2

1

1=

e

e and

cos2= .

Definition 1.2 A function A)(zf is in the class ,,,,dkQm ,if and only if

2

1,,,,1

>1,,,,1

1

2

zdkmzdkmeRe i

JJ ,

where

.)(),(

)(),(

=,,,,zfdkmD

'zfdkmDz

zdkm

J (1.14)

Since sincos)(=)( izhezKi is a convex univalent function, we can write

Definition 1.1 in subordination form

).()(,,,,,,,, UzzKzdkmandfdkQf m JA

Special Cases

For 0= and ,,,,dkQfm we have ,,)(),(

*SzfdkDm .

2 PRELIMINARY RESULTS

Lemma 1 [13] Let h be a convex univalent function in U with 0>)( zhRe , where Uz {0},\, CC . If p is analytic in U with (0)=(0) hp , then

).()(

)()( zh

zp

z'zpzp


Lemma 2 [12] Let d be a complex number with 0>dRe . Suppose that CC 2:

is continuous and satisfies the conditions 0),( yixRe , when x is real and

).)/(2||( 2 dReixdy

If p is analytic in U with dp =(0) and 0>)(),(

z'zpzpRe for Uz ,

then 0>)(zpRe in U .

Lemma 3 [18] Let f and g be in the class Vk and *S respectively. Then, for

every function F analytic in U , we have

,,)()()(

)()()(UzUFco

zgzf

zFzgzf

where )(UFco , denotes the closed convex hull of the set )(UF .

Lemma 4 [16] Let the function )(z given by

nnn

zBz

1=

=)(

be convex in U . Suppose also that the function )(zh given by

nnn

zhzh

1=

=)(

is holomorphic in U . If )()( zzh , Uz , then 1,2,3,=|,||| 1 NnBhn .

3 COEFFICIENT INEQUALITIES

A function A)(zf is said to be bi-univalent in U if both )(zf and )(1 zf are

univalent in U . Let denote the class of bi-univalent functions defined in the unit disk U .

Definition 3.1 Let CUh : be a convex univalent function such that 1=(0)h and

)(=)( zhzh , for Uz and 0>))(( zhRe . A function f is said to be in the class


{0}\),,( CS

if the following conditions are satisfied:

Uzizhzf

zzfef

'i

,sincos)(1

)(

)(11,

(3.1)

and

,,sincos)(1)(

)(11 Uih

g

zge

'i

(3.2)

where

2,

2,)5(5)(2=)(=)( 4432

3

2

3

3

2

2

2

2

1 aaaaaaafg .

Definition 3.2 A function f given by (1.1) is said to be in the class

),,,,(, hdkm

S if the following conditions are satisfied:

UzizhzfdkmD

'zfdkmDz

ie

,s inco s)(1)(),(

)(),(1

1

(3.3)

and

,,s inco s)(1)(),(

)(),(1

1 UihgdkmD

'gdkmD

ie

(3.4)

where Uzfg ,1,{0},0\),(=)( 1 C on specializing the parameter .

Theorem 1 Let f given by (1.1) be in the class ),(

S , then

cos|||||| 12 Ba

and

.cos||||4|| 13 Ba


Theorem 2 Let f given by (1.1) be in the class ),,,,(, hdkm

S , then

2

1

22

2

2

12

)8(

)(1

)8(

)2(1

cos||||2||

k

d

k

d

Ba

mm

and

.

)8(

)(1

)8(

)2(1

cos||||2||

2

1

22

2

2

1

2

3

k

d

k

d

Ba

mm

Proof. It follows from (3.3) and (3.4) that

UzizpzfdkmD

'zfdkmDz

ie

,sincos)(=1)(),(

)(),(1

1

(3.5)

and

,,sincos)(=1)(),(

)(),(1

1 UiqgdkmD

'gdkmD

ie

(3.6)

where )()( zhzp and )()( hq have the forms

2211=)( zpzpzp

and

2211=)( qqq

respectively. It follows from (3.5) and (3.6) that

cos=)(4(1)!

)()(112

1

pak

de mi (3.7)

cos=)(4(1)!

)()(1

)((2)!4

)2(2)(12

2

2

2

1

3

2

2

2

pak

da

k

de mmi

(3.8)

cos=)(4(1)!

)()(112

1

qak

de mi (3.9)


and

.cos=)(4(1)!

)()(1

)((2)!4

)2(2)(1

)((2)!4

)2(4)(12

2

2

2

1

3

2

2

22

2

2

2

2

qak

da

k

da

k

de mmmi

(3.10)

From (3.7) and (3.9), we obtain

.= 11 qp

Adding (3.8) and (3.10), we get

.cos)(=)8(

)(1

)8(

)2(122

2

22

1

22

2

2

qpak

d

k

de mmi

(3.11)

Since )(, Uhqp , applying Lemma 4, we have

NmBm

pp

m

m ,!

(0)= 1

)(

(3.12)

NmBm

qq

m

m ,!

(0)= 1

)(

. (3.13)

Applying (3.12), (3.13) and Lemma 4 for the coefficients 121 ,, qpp and 2q , we get

21)8(

22)(1

2)8(

2)2(1

cos|1|||2|2|

k

dm

k

dm

Ba

. (3.14)

Subtracting (3.10) from (3.8), we get

cos)(=)8(

)2(1

)8(

)2(122

2

2

2

2

3

2

2

qpak

da

k

de mmi

(3.15)

or, equivalently

21)8(

22)(1

2)8(

2)2(1

cos)22(2

2)2(1

cos)22(2)8(=3

k

dm

k

dm

qp

iedm

qpk

iea

.


Applying (3.12), (3.13) and Lemma 4 once again for the coefficients 121 ,, qpp and 2q, we get

.

)8(

)(1

)8(

)2(1

cos||||2||

2

1

22

2

2

1

2

3

k

d

k

d

Ba

mm

This complete the proof of Theorem 2.

Corollary 1 Let Af be bi convex function of order then 1|| 2a and

1|| 3a .

Corollary 2 Let Af satisfy the condition )()(

)(1 zh

z'f

z''zf and

)()(

)(1 wh

w'g

w''zg then

|22|2||

21

2

1

11

2

BBB

BBa

and |)12|(

2

1|| 13 BBBa .

Corollary 3 Let f be given by (1.1) and 1= fg . If f and g satisfies the

condition )()(

)(1

)(

)()(1 zh

z'f

z''zf

zf

z'zf

and )()(

)(1

)(

)()(1 wg

w'g

w''wg

wg

w'wg

,

then

|))((1|)(1

||

21

2

1

11

2

BBB

BBa

(3.16)

and

.1

|12||| 13

BBBa (3.17)


Remark 2 Put 1= in corollary 3, we get the result |22|2

||

21

2

1

11

2

BBB

BBa

and

|)12|(2

1|| 13 BBBa of corollary 2.

4 CLOSURE PROPERTY

Theorem 3 Let

).(),(=))(,,,,( zfdkDzfdkmF m (4.1)

Then ,,,,dkQfm if and only if ,,))(,,,,( *SzfdkmF .

Proof. Let ,,))(,,,,( *SzfdkmF , then

.

2

1))(,,,,(

)))((,,,,(1>

2

1))(,,,,(

)))((,,,,(11

zfdkmF

zzfdkm'zF

zfdkmF

zzfdkm'zFieRe

(4.2)

Thus (4.1) together with (4.2) implies

,2

1),,,,(1

>

2

1),,,,(1

1

zdkmzdkmieRe

JJ

where ),,,,( zdkmJ is given by (1.14). Therefore ,,,,)( dkQzfm .

Converse is immediate.

Theorem 4 For 1m , ,,,,,,,,1 dkQdkQ mm .

Proof. Let ,,,,1 dkQf m and

)(=)(),(

)(),(

zpzfdkmD

'zfdkmDz

, (4.3)

where p is analytic in U and 1=(0)p .


From (1.12) and (4.3) and after some simplification, we obtain

).(1)(=)(),(

)(),(1

zpzfdkD

zfdkDm

m

(4.4)

By logarithmic differentiation of (4.4), we have

.)()(1

)()(=

)(),(1

)(),(1

zp

z'zpzp

zfdkmD

'zfdkmDz

(4.5)

Since ,,,,1 dkQf m , so

).()()(1

)()( zK

zp

z'zpzp

Thus by using Lemma 1, )()( zKzp and hence ,,,,dkQfm .

Let us consider the Bernardi integral operator F given by

.)(1=)( 10

dttftz

zfFz

(4.6)

For with 1> Re the operator has the property AA :F , (see, for instance,

[10], p.11).

Theorem 5 Let 1> and ,,,,dkQfm , then ,,,,dkQfF

m .

Proof. Let ,,,,dkQfm and

)(=))((),(

))((),(

zpzfFdkmD

'zfFdkmDz

, (4.7)

where the function )(zp is analytic in U and 1=(0)p . From (4.6), we have

)(1)(=))(()()( zfzfFz'fFz

and so

))((),()(),(1)((=))((),( zfFdkmDzfdkmD'zfFdkmDz

. (4.8)


Then, by using equations (4.7) and (4.8), we obtain

)(=))((),(

)(),(1)((zp

zfFdkD

zfdkDm

m

(4.9)

By logarithmic differentiation of(4.9), we have

).(,)(

)(=)(),(

)(),(

zzp

'zpzp

zfdkmD

'zfdkmDz

Hence by Lemma1, we conclude that )()( zKzp in U which implies that

,,,,dkQfFm .

Theorem 6 Let ,,,,dkQfm and Vk . If 1


By Theorem(3), the function ,,)(),(=)(*SzfdkDzF m .

Hence, by Lemma (3) , we have

).()]([)(),(

)(),(

zKUFcozFdkmD

'zFdkmDz

Since )(zK is a convex univalent and )()( zKzp .

Hence ,,,,= dkQfFm .

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