Certain Subclasses of Starlike Functions of Complex Order...

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2010, Article ID 178605, 12 pagesdoi:10.1155/2010/178605

Research ArticleCertain Subclasses of StarlikeFunctions of Complex Order Involving GeneralizedHypergeometric Functions

G. Murugusundaramoorthy1 and N. Magesh2

1 School of Advanced Sciences, VIT University, Vellore 632014, India2 Department of Mathematics, Government Arts College (Men), Krishnagiri 635001, India

Correspondence should be addressed to G. Murugusundaramoorthy, [email protected]

Received 24 November 2009; Accepted 18 March 2010

Academic Editor: Stanisława R. Kanas

Copyright q 2010 G. Murugusundaramoorthy and N. Magesh. This is an open access articledistributed under the Creative Commons Attribution License, which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited.

Making use of the generalized hypergeometric functions, we define a new subclass of uniformlyconvex functions and a corresponding subclass of starlike functions with negative coefficientsand obtain coefficient estimates, extreme points, the radii of close-to-convexity, starlikeness andconvexity, and neighborhood results for the class TSl

m(α, β, γ). In particular, we obtain integralmeans inequalities for the function f that belongs to the class TSl

m(α, β, γ) in the unit disc.

1. Introduction

Let A denote the class of functions of the form

f(z) = z +∞∑

n=2

anzn (1.1)

which are analytic in the open unit disc U = {z : z ∈ C, |z| < 1} and normalized by f(0) =f ′(0) − 1 = 0. Also denote by T the subclass of A consisting of functions of the form

f(z) = z −∞∑

n=2|an|zn, (1.2)

introduced and studied by Silverman [1]. For functions f ∈ A given by (1.1) and g ∈ Agiven by g(z) = z +

∑∞n=2 bnz

n, it is known [2] the definition of the Hadamard product

2 International Journal of Mathematics and Mathematical Sciences

(or Convolution) of f and g by

(f ∗ g)(z) = z +

∞∑

n=2

anbnzn, z ∈ U. (1.3)

For positive real values of α1, . . . , αl and β1, . . . , βm (βj /= 0,−1, . . . ; j = 1, 2, . . . , m) thegeneralized hypergeometric function lFm(z) is defined by

lFm(z) ≡ lFm

(α1, . . . , αl; β1, . . . , βm; z

):=

∞∑

n=0

(α1)n · · · (αl)n(β1)n · · ·

(βm)n

zn

n!

(l ≤ m + 1; l,m ∈ N0 := N ∪ {0}; z ∈ U),

(1.4)

where N denotes the set of all positive integers and (λ)k is the Pochhammer symbol definedby

(λ)n =

⎧⎨

⎩1, n = 0,

λ(λ + 1)(λ + 2) · · · (λ + n − 1), n ∈ N.(1.5)

The notation lFm is quite useful for representing many well-known functions such as theexponential, the Binomial, the Bessel, the Laguerre polynomial, and others; for example, see[3].

Let H(α1, . . . , αl; β1, . . . , βm) : A → A be a linear operator defined by

H(α1, . . . , αl; β1, . . . , βm

)f(z) := zlFm

(α1, α2, . . . , αl; β1, β2, . . . , βm; z

) ∗ f(z)

= z +∞∑

n=2

Γnanzn,

(1.6)

where

Γn =(α1)n−1 · · · (αl)n−1(β1)n−1 · · ·

(βm)n−1

1(n − 1)!

, (1.7)

unless otherwise stated. For notational simplicity, we can use a shorter notation Hlm[α1] for

H(α1, . . . , αl; β1, . . . , βm) in the sequel. The linear operator Hlm[α1] is called Dziok-Srivastava

operator (see [3]), includ (as its special cases) various other linear operators introduced andstudied by Carlson and Shaffer [4], Owa [5], Ruscheweyh [2], and Srivastava and Owa[6]. Motivated by Goodman [7, 8], Rønning [9, 10] introduced and studied the followingsubclasses of A. A function f ∈ A is said to be in the class Sp(α, β), uniformly β-starlikefunctions if it satisfies the condition

Re{zf ′(z)f(z)

− α

}> β

∣∣∣∣zf ′(z)f(z)

− 1∣∣∣∣,(−1 < α ≤ 1; β ≥ 0

)z ∈ U (1.8)

International Journal of Mathematics and Mathematical Sciences 3

and is said to be in the class UCV(α, β), uniformly β-convex functions if it satisfies thecondition

Re{1 +

zf ′′(z)f ′(z)

− α

}> β

∣∣∣∣zf ′′(z)f ′(z)

∣∣∣∣,(−1 < α ≤ 1; β ≥ 0

)z ∈ U. (1.9)

Indeed it follows from (1.8) and (1.9) that

f ∈ UCV(α, β)⇐⇒ zf ′ ∈ Sp

(α, β). (1.10)

The interesting geometric properties of these function classes were extensively studied byKanas et al., in [11–14]. Motivated by Altintas et al. [15], Murugusundaramoorthy andSrivastava [16], and Murugusundaramoorthy and Magesh [17], now, we define a newsubclass uniformly starlike functions of complex order.

For −1 ≤ α < 1, β ≥ 0, and γ ∈ C \ {0}, we let TSlm(α, β, γ) be the class of functions f

satisfying (1.2) with the analytic criterion

Re

{1 +

1γ

(z(Hl

m[α1]f(z))′

Hlm[α1]f(z)

− α

)}> β

∣∣∣∣∣1 +1γ

(z(Hl

m[α1]f(z))′

Hlm[α1]f(z)

− 1

)∣∣∣∣∣, z ∈ U (1.11)

where Hlm[α1]f(z) is given by (1.6).

By suitably specializing the values of l, m, α1, α2, . . . , αl, β1, β2, . . . , βm, α, γ , and β theclass TSl

m(α, β, γ), leads to various new subclasses of starlike functions of complex order. Asfor illustrations, we present some examples for the cases.

Example 1.1. If l = 2, m = 1 with α1 = 1, α2 = 1, β1 = 1, and f of the form of (1.2), then

TS(α, β, γ

):={Re{1 +

1γ

(zf ′(z)f(z)

− α

)}> β

∣∣∣∣1 +1γ

(zf ′(z)f(z)

− 1)∣∣∣∣, z ∈ U

}. (1.12)

Example 1.2. If l = 2, m = 1 with α1 = δ + 1 (δ > −1), α2 = 1, β1 = 1, and f of the form of (1.2),then

TRδ

(α, β, γ

):=

{Re

{1 +

1γ

(z(Dδf(z)

)′

Dδf(z)− α

)}> β

∣∣∣∣∣1 +1γ

(z(Dδf(z)

)′

Dδf(z)− 1

)∣∣∣∣∣, z ∈ U

},

(1.13)

where Dδ is called Ruscheweyh derivative of order δ (δ > −1) defined by

Dδf(z) :=z

(1 − z)δ+1∗ f(z) ≡ H2

1(δ + 1, 1; 1)f(z). (1.14)


Example 1.3. If l = 2, m = 1 with α1 = μ + 1 (μ > −1), α2 = 1, β1 = μ + 2, and f of the form of(1.2), then

TBμ

(α, β, γ

)=

{Re

{1 +

1γ

(z(Jμf(z)

)′

Jμf(z)− α

)}> β

∣∣∣∣∣1 +1γ

(z(Jμf(z)

)′

Jμf(z)− 1

)∣∣∣∣∣, z ∈ U

},

(1.15)

where Jμ is a Bernardi operator [18] defined by

Jμf(z) :=μ + 1zμ

∫z

0tμ−1f(t)dt ≡ H2

1

(μ + 1, 1;μ + 2

)f(z). (1.16)

Example 1.4. If l = 2, m = 1 with α1 = a (a > 0), α2 = 1, β1 = c (c > 0), and f of the form of(1.2), then

TLac

(α, β, γ

):=

{Re

{1 +

1γ

(z(L(a, c)f(z)

)′

L(a, c)f(z)− α

)}

> β

∣∣∣∣∣1 +1γ

(z(L(a, c)f(z)

)′

L(a, c)f(z)− 1

)∣∣∣∣∣, z ∈ U

},

(1.17)

where L(a, c) is a well-known Carlson-Shaffer linear operator [4] defined by

L(a, c)f(z) :=

( ∞∑

k=0

(a)k(c)k

zk+1)

∗ f(z) ≡ H21(a, 1; c)f(z). (1.18)

Also TLac (α, β, γ) was studied by Murugusundaramoorthy and Magesh [19].

The main object of this paper is to study some usual properties of the geometricfunction theory such as the coefficient bound, extreme points, radii of close to convexity,starlikeness, and convexity for the class TSl

m(α, β, γ). Further, we obtain neighborhood resultsand integral means inequalities for aforementioned class.

2. Basic Properties

First we obtain the necessary and sufficient condition for functions f in the class TSlm(α, β, γ).

Theorem 2.1. The necessary and sufficient condition for f of the form of (1.2) to be in the classTSl

m(α, β, γ) is

∞∑

n=2

[(n +∣∣γ∣∣)(1 − β

) − (α − β)]Γn|an| ≤ (1 − α) +

∣∣γ∣∣(1 − β

), (2.1)

where −1 ≤ α < 1, β ≥ 0, and γ ∈ C \ {0}.


Proof. Assume that f ∈ TSlm(α, β, γ), then

Re

{1 +

1γ

(z(Hl

m[α1]f(z))′

Hlm[α1]f(z)

− α

)}> β

∣∣∣∣∣1 +1γ

(z(Hl

m[α1]f(z))′

Hlm[α1]f(z)

− 1

)∣∣∣∣∣ ,

Re{1 +

1γ

(z(1 − α) −∑∞

n=2(n − α)Γn|an|znz −∑∞

n=2 Γn|an|zn)}

> β

∣∣∣∣1 −1γ

(∑∞n=2(n − 1)Γn|an|znz −∑∞

n=2 Γn|an|zn)∣∣∣∣ .

(2.2)

Letting z → 1− along the real axis, we have

{1 +

1∣∣γ∣∣

((1 − α) −∑∞

n=2(n − α)Γn|an|1 −∑∞

n=2 Γn|an|)}

> β

[1 − 1∣∣γ

∣∣

(∑∞n=2(n − 1)Γn|an|1 −∑∞

n=2 Γn|an|)]

. (2.3)

Hence, by maximum modulus theorem, the simple computational leads the desiredinequality

∞∑

n=2

[(n +∣∣γ∣∣)(1 − β

) − (α − β)]Γn|an| ≤ (1 − α) +

∣∣γ∣∣(1 − β

). (2.4)

Conversely, suppose that (2.1) is true for z ∈ U. Then

Re

{1 +

1γ

(z(Hl

m[α1]f(z))′

Hlm[α1]f(z)

− α

)}− β

∣∣∣∣∣1 +1γ

(z(Hl

m[α1]f(z))′

Hlm[α1]f(z)

− 1

)∣∣∣∣∣ > 0 (2.5)

if

1 +1∣∣γ∣∣

((1 − α) −∑∞

n=2(n − α)Γn|an||z|n−11 −∑∞

n=2 Γn|an||z|n−1)

− β

[1 − 1∣∣γ

∣∣

(∑∞n=2(n − 1)Γn|an||z|n−11 −∑∞

n=2 Γn|an||z|n−1)]

> 0.

(2.6)

That is if

∞∑

n=2

[(n +∣∣γ∣∣)(1 − β

) − (α − β)]Γn|an| ≤ (1 − α) +

∣∣γ∣∣(1 − β

), (2.7)

which completes the proof.

Corollary 2.2. Let the function f defined by (1.2) belong to TSlm(α, β, γ). Then

|an| ≤[(1 − α) +

∣∣γ∣∣(1 − β

)][(n +∣∣γ∣∣)(1 − β

) − (α − β)]Γn

, (2.8)


n ≥ 2, −1 ≤ α < 1, β ≥ 0 and, γ ∈ C \ {0}, with equality for

f(z) = z −[(1 − α) +

∣∣γ∣∣(1 − β

)][(n +∣∣γ∣∣)(1 − β

) − (α − β)]Γn

zn. (2.9)

Next we state the following theorem on extreme points for the class TSlm(α, β, γ)

without proof.

Theorem 2.3 (Extreme Points). Let

f1(z) = z, fn(z) = z −[(1 − α) +

∣∣γ∣∣(1 − β

)][(n +∣∣γ∣∣)(1 − β

) − (α − β)]Γn

zn for n = 2, 3, 4, . . . . (2.10)

Then f ∈ TSlm(α, β, γ) if and only if f can be expressed in the form f(z) =

∑∞n=1 λnfn(z), where

λn ≥ 0 and∑∞

n=1 λn = 1.

3. Close-to-Convexity, Starlikeness, and Convexity

We determine the radii of close-to-convexity, starlikeness, and convexity results for functionsin the class TSl

m(α, β, γ) in the following theorems.

Theorem 3.1. Let f ∈ TSlm(α, β, γ). Then f is close-to-convex of order δ (0 ≤ δ < 1) in the disc

|z| < r1, where

r1 = infn≥2

[(1 − δ)

n

[(n +∣∣γ∣∣)(1 − β

) − (α − β)]

[(1 − α) +

∣∣γ∣∣(1 − β

)] Γn

]1/(n−1). (3.1)

Proof. Let f belong to T. It is known [20] that f is close-to-convex of order δ, if it satisfies thecondition

∣∣f ′(z) − 1∣∣ < 1 − δ. (3.2)

For the left-hand side of (3.2) we have

∣∣f ′(z) − 1∣∣ ≤

∞∑

n=2

n|an||z|n−1. (3.3)

The last expression is less than 1 − δ if

∞∑

n=2

n

1 − δ|an||z|n−1 < 1. (3.4)


Using the fact that f ∈ TSlm(α, β, γ) if and only if

∞∑

n=2

[(n +∣∣γ∣∣)(1 − β

) − (α − β)]

(1 − α) +∣∣γ∣∣(1 − β

) Γn|an| < 1. (3.5)

we can say that (3.2) is true if

n

1 − δ|z|n−1 ≤

[(n +∣∣γ∣∣)(1 − β

) − (α − β)]

(1 − α) +∣∣γ∣∣(1 − β

) Γn. (3.6)

Or, equivalently,

|z| ≤[(1 − δ)

[(n +∣∣γ∣∣)(1 − β

) − (α − β)]

n[(1 − α) +

∣∣γ∣∣(1 − β

)] Γn

]1/(n−1), (3.7)

which completes the proof.

Theorem 3.2. Let f ∈ TSlm(α, β, γ). Then the following are given.

(1) f is starlike of order δ (0 ≤ δ < 1) in the disc |z| < r2, where

r2 = infn≥2

{(1 − δ)(n − δ)

[(n +∣∣γ∣∣)(1 − β) − (α − β)]

[(1 − α) +∣∣γ∣∣(1 − β)]

Γn

}1/(n−1). (3.8)

(2) f is convex of order δ (0 ≤ δ < 1) in the unit disc |z| < r3, where

r3 = infn≥2

{(1 − δ)n(n − δ)

[(n +∣∣γ∣∣)(1 − β

) − (α − β)]

[(1 − α) +

∣∣γ∣∣(1 − β

)] Γn

}1/(n−1). (3.9)

Each of these results is sharp for the extremal function f given by (2.10).

Proof. Let f ∈ T. It is known [1] that f is starlike of order δ, if it satisfies the condition

∣∣∣∣zf ′(z)f(z)

− 1∣∣∣∣ < 1 − δ. (3.10)

For the left-hand side of (3.10)we have

∣∣∣∣zf ′(z)f(z)

− 1∣∣∣∣ ≤∑∞

n=2(n − 1)|an||z|n−11 −∑∞

n=2|an||z|n−1. (3.11)


The last expression is less than 1 − δ if

∞∑

n=2

n − δ

1 − δ|an||z|n−1 < 1. (3.12)

Using the fact that f ∈ TSlm(α, β, γ) if and only if

∞∑

n=2

[(n +∣∣γ∣∣)(1 − β

) − (α − β)]

(1 − α) +∣∣γ∣∣(1 − β

) Γn|an| < 1. (3.13)

we can say that (3.10) is true if

n − δ

1 − δ|z|n−1 <

[(n +∣∣γ∣∣)(1 − β

) − (α − β)]

(1 − α) +∣∣γ∣∣(1 − β

) Γn. (3.14)

Or, equivalently,

|z|n−1 < (1 − δ)[(n +∣∣γ∣∣)(1 − β

) − (α − β)]

(n − δ)[(1 − α) +

∣∣γ∣∣(1 − β

)] Γn, (3.15)

which yields the starlikeness of the family.Using the fact that f is convex if and only if zf ′ is starlike, we can prove (2), on lines

similar to those the proof of (1).

4. Integral Means

In order to find the integral means inequality and to verify the Silverman Conjuncture [21]for f ∈ TSl

m(α, β, γ) we need the following subordination result due to Littlewood [22].

Lemma 4.1 (see [22]). If the functions f and g are analytic inU with g ≺ f , then

∫2π

0

∣∣∣g(reiθ)∣∣∣

ηdθ ≤

∫2π

0

∣∣∣f(reiθ)∣∣∣

ηdθ, η > 0, z = reiθ, 0 < r < 1. (4.1)

Applying Theorem 2.1 with the extremal function and Lemma 4.1, we prove thefollowing theorem.

Theorem 4.2. Let η > 0. If f ∈ TSlm(α, β, γ) and {Φ(α, β, γ, n)}∞n=2 is nondecreasing sequence, then,for z = reiθ and 0 < r < 1, one has

∫2π

0

∣∣∣f(reiθ)∣∣∣ηdθ ≤

∫2π

0

∣∣∣f2(reiθ)∣∣∣ηdθ, (4.2)

where f2(z) = z−((1−α)+|γ |(1−β))z2/Φ(α, β, γ, 2), andΦ(α, β, γ, n) = [(n+|γ |)(1−β)−(α−β)]Γn.


Proof. Let f of the form of (1.2) and f2(z) = z − ((1 − α) + |γ |(1 − β))z2/Φ(α, β, γ, 2), then wemust show that

∫2π

0

∣∣∣∣∣1 −∞∑

n=2

anzn−1∣∣∣∣∣

η

dθ ≤∫2π

0

∣∣∣∣∣1 −(1 − α) +

∣∣γ∣∣(1 − β)

Φ(α, β, γ, 2)z

∣∣∣∣∣

η

dθ. (4.3)

By Lemma 4.1, it suffices to show that

1 −∞∑

n=2|an|zn−1 ≺ 1 − (1 − α) +

∣∣γ∣∣(1 − β

)

Φ(α, β, γ, 2

) z. (4.4)

Setting

1 −∞∑

n=2|an|zn−1 = 1 − (1 − α) +

∣∣γ∣∣(1 − β

)

Φ(α, β, γ, 2

) w(z), (4.5)

from (4.5) and (2.1)we obtain

|w(z)| =∣∣∣∣∣

∞∑

n=2

Φ(α, β, γ, n

)

(1 − α) +∣∣γ∣∣(1 − β

)anzn−1∣∣∣∣∣

≤ |z|∞∑

n=2

Φ(α, β, γ, n

)

(1 − α) +∣∣γ∣∣(1 − β

) |an|

≤ |z| < 1.

(4.6)

This completes the proof of Theorem 4.2.

5. Inclusion Relations Involving Nδ(e)

To study about the inclusion relations involvingNδ(e)we need the following definitions dueto Goodman [23] and Ruscheweyh [24]. The n, δ neighborhood of function f ∈ T is given by

Nδ

(f)=

{g ∈ T : g(z) = z −

∞∑

n=2|bn|zn,

∞∑

n=2

n|an − bn| ≤ δ

}. (5.1)

Particularly for the identity function e(z) = z, we have

Nδ(e) =

{g ∈ T : g(z) = z −

∞∑

n=2|bn|zn,

∞∑

n=2

n|bn| ≤ δ

}. (5.2)


Theorem 5.1. Let

δ =2[(1 − α) +

∣∣γ∣∣(1 − β

)][(2 +∣∣γ∣∣)(1 − β

) − (α − β)]Γ2

, (5.3)

where Γ2 = (α1 · · ·α2)/(β1 · · · β2). Then TSlm(α, β, γ) ⊂ Nδ(e).

Proof. For f ∈ TSlm(α, β, γ), Theorem 2.1 yields

[(2 +∣∣γ∣∣)(1 − β

) − (α − β)]Γ2

∞∑

n=2|an| ≤ (1 − α) +

∣∣γ∣∣(1 − β

)(5.4)

so that

∞∑

n=2|an| ≤

(1 − α) +∣∣γ∣∣(1 − β

)[(2 +∣∣γ∣∣)(1 − β

) − (α − β)]Γ2

. (5.5)

On the other hand, from (2.1) and (5.5) we have

(1 − β

)Γ2

∞∑

n=2

n|an| ≤ (1 − α) +∣∣γ∣∣(1 − β

)+[(α − β

) − ∣∣γ∣∣(1 − β)]Γ2

∞∑

n=2|an|

≤ (1 − α) +∣∣γ∣∣(1 − β

)+[(α − β

) − ∣∣γ∣∣(1 − β)]

× Γ2(1 − α) +

∣∣γ∣∣(1 − β

)[(2 +∣∣γ∣∣)(1 − β

) − (α − β)]Γ2

≤[(1 − α) +

∣∣γ∣∣(1 − β

)]2(1 − β

)[(2 +∣∣γ∣∣)(1 − β

) − (α − β)] ,

∞∑

n=2

n|an| ≤2[(1 − α) +

∣∣γ∣∣(1 − β

)][(2 +∣∣γ∣∣)(1 − β

) − (α − β)]Γ2

.

(5.6)

Now we determine the neighborhood for each of the class TSlm(α, β, γ) which we

define as follows. A function f ∈ T is said to be in the class TSlm(α, β, γ) if there exists a

function g ∈ TSlm(α, β, γ) such that

∣∣∣∣f(z)g(z)

− 1∣∣∣∣ < 1 − η,

(z ∈ U, 0 ≤ η < 1

). (5.7)

Theorem 5.2. If g ∈ TSlm(α, β, γ) and

η = 1 − δ[(2 +∣∣γ∣∣)(1 − β

) − (α − β)]Γ2

2[((

2 +∣∣γ∣∣)(1 − β

) − (α − β))Γ2 −

((1 − α) +

∣∣γ∣∣(1 − β

))] , (5.8)

where Γ2 = (α1 · · ·α2)/(β1 · · · β2), thenNδ(g) ⊂ TSlm(α, β, γ, η).


Proof. Suppose that f ∈ Nδ(g), then we find from (5.6) that

∞∑

n=2|an − bn| ≤ δ, (5.9)

which implies the coefficient inequality

∞∑

n=2|an − bn| ≤ δ

2. (5.10)

Next, since g ∈ TSlm(α, β, γ), we have

∞∑

n=2|bn| ≤

2[(1 − α) +

∣∣γ∣∣(1 − β

)][(2 +∣∣γ∣∣)(1 − β

) − (α − β)]Γ2

(5.11)

so that

∣∣∣∣f(z)g(z)

− 1∣∣∣∣ <∑∞

n=2|an − bn|1 −∑∞

n=2|bn|

≤ δ

2×

[(2 +∣∣γ∣∣)(1 − β

) − (α − β)]Γ2[((

2 +∣∣γ∣∣)(1 − β

) − (α − β))Γ2 −

((1 − α) +

∣∣γ∣∣(1 − β

))]

≤ 1 − η,

(5.12)

provided that η is given precisely by (5.8). Thus by definition, f ∈ TSlm(α, β, γ, η) for η given

by (5.8), which completes the proof.

Concluding Remarks

By suitably specializing the various parameters involved in Theorems 2.1 to 5.2, we can statethe corresponding results for the new subclasses defined in Examples 1.1 to 1.4 and also formany relatively more familiar function classes.

Acknowledgment

The authors would like to record their sincerest thanks to the referee(s) for their valuablesuggestions and comments.

References

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