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International Mathematical Forum, 5, 2010, no. 37, 1839 - 1848

A Subclass of Uniformly Convex Functions and

a Corresponding Subclass of Starlike Functions

with Fixed Second Coefficient Defined by

Carlson and Shaffer Operator

S. M. Khairnar

Department of Mathematics

Maharashtra Academy of Engineering

Alandi, Pune, Maharashtra, India

smkhairnar2007@gmail.com

N. H. More

Department of Applied Sciences

Rajiv Gandhi Institute of Technology

Andheri (West), Mumbai - 53, Maharashtra, India

nhmore@rediffmail.com

Abstract

The main objective of this paper is to obtain necessary and sufficient

condition for a subclass of uniformly convex functions and correspond-

ing subclass of starlike functions with fixed second coefficient defined

by Carlson and Shaffer operator for the function f(z) in UCT(α,β).

Furthermore, we obtain extreme points, distortion bounds and closure

properties for f(z) in UCT(α,β) by fixing second coefficient.

Mathematics Subject Classification: 30C45

Keywords: Uniformly convex function, Convex function, Carlson and

Shaffer operator, Hadamard product, distortion bounds and extreme points

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S. M. Khairnar and N. H. More

1. Introduction

Denote by S the class of functions of the form

f(z) = z +

∞

?

n=2

anzn

(1.1)

that are analytic and univalent in the unit disc U = {z : |z| < 1} and by

ST and CV the subclasses of S that are respectively, starlike and convex.

Goodman [4, 5] introduced and defined the following subclasses of CV and

ST.

A function f(z) is uniformly convex (uniformly starlike) in U if f(z) is in

CV (ST) and has the property that for every circular arc γ contained in U,

with center ξ also in U, the arc f(γ) is convex (starlike) with respect to f(ξ).

The class of uniformly convex functions is denoted by UCV and the class of

uniformly starlike functions by UST. It is well known from [[3], [8]] that

????

In [10], Rønning introduced a new class of starlike fucntions related to UCV

defined as

????

Spby introducing a parameter α,−1 ≤ α < 1,

????

Now we define the function φ(a,c;z) by

f ∈ UCV ⇔

zf??(z)

f?(z)

????≤ Re

????≤

?zf??(z)

f?(z)

?

.

f ∈ Sp⇔

zf?(z)

f(z)

− 1

?zf?(z)

f(z)

?

.

Note that f(z) ∈ UCV ⇔ zf?(z) ∈ Sp. Further, Rønning generalized the class

f ∈ Sp(α) ⇔

zf?(z)

f(z)

− 1

????≤ Re

?zf?(z)

f(z)

− α

?

.

φ(a,c;z) = z +

∞

?

n=2

(a)n−1

(c)n−1zn, (1.2)

for c ?= 0,−1,−2,···,a ?= −1;z ∈ U where (λ)nis the Pochhammer symbol

defined by

(λ)n =

Γ(n + λ)

(Γ(λ)

?1;

=

n = 0

λ(λ + 1)(λ + 2)···(λ + n − 1), n ∈= {1,2,···}

?

(1.3)

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Subclass of uniformly convex functions

1841

Carlson and Shaffer [3] introduced a linear operator L(a,c), defined by

L(a,c)f(z) = φ(a,c;z) ∗ f(z)

= z +

∞

?

n=2

(a)n−1

(c)n−1anzn, z ∈ U,(1.4)

where ∗ stands for the Hadamard product or convolution product of two power

series

∞

?

defined by

ϕ(z) =

n=1

ϕnzn

and ψ() =

∞

?

n=1

ψnzn

(ϕ ∗ ψ)(z) = ϕ(z) ∗ ψ(z) =

∞

?

n=1

ϕnψnzn.

We note that L(a,a)f(z) = f(z),L(2,1)f(z) = zf;(z),L(m + 1,1)f(z) =

Dmf(z), where Dmf(z) is the Ruscheweyh derivative of f(z) defined by Ruscheweyh

[11] as

Dmf(z) =

(1 − z)m+1∗ f(z),

which is equivalently,

z

m > −1, (1.5)

Dmf(z) =

z

m!

dm

dzm{zm−1f(z)}.

Definition 1.1 : For β ≥ 0,−1 ≤ α < 1, we define a class UCV (α,β) subclass

of S consisting of functions f(z) of the form (1.1) and satisfying the analytic

criterion

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

Re

(L(a,c)f(z))?+ 1 − α

?

≥ β

????

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

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

????, z ∈ U. (1.6)

We also let UCT(α,β), the subclass of S consisting of functions of the form

f(z) = z −

∞

?

n=2

anzn, an≥ 0, ∀ n ≥ 2. (1.7)

The main objective of this paper is to obtain necessary and sufficient condi-

tion for a subclass of uniformly convex functions and corresponding subclass of

starlike functions with fixed second coefficient defined by Carlson and Shaffer

operator for the function f(z) ∈ UCT(α,β). Furthermore, we obtain extreme

points, distortion bounds and closure properties for f(z) ∈ UCT(α,β) by

fixing second coefficient.

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S. M. Khairnar and N. H. More

2. The Class UCT(α,β)

Firstly, we obtain necessary and sufficient condition for functions f(z) in

the calsses UCV (α,β).

Theorem 2.1 : A function f(z) of the form (1.1) is in UCV (α,β) if

∞

?

n=2

n[n(1 + β) − (α + β)](a)n−1

(c)n−1|an≤ 1 − α, (2.1)

−1 ≤ α < 1,β ≥ 0.

Proof : If suffices to show that

????

β

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

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

????− Re

????− Re

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

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

?

≤ 1 − α.

We have

β

????

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

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

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

(c)n−1|an|

.

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

?

≤ 1 − α,

≤

(1 + β)

∞

?

n=2n(n − 1)(a)n−1

∞

?

1 −

n=2n(a)n−1

(c)n−1|an|

The last expression is bounded above by (1 − α) if

∞

?

and hence the proof is complete.

Theorem 2.2 : a necessary and sufficient for f(z) of the form (1.7) to be in

the class UCT(α,β),−1 ≤ α < 1,β ≥ 0 is that

∞

?

Proof : In view of Theorem 2.1, we need only to prove the necessity. If

f(z) ∈ UCT(α,β) and z is a real then

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

which gives

n=2

n[n(1 + β) − (α + β)](a)n−1

(c)n−1|an| ≤ 1 − α,

n=2

n[n(1 + β) − (α + β)](a)n−1

(c)n−1an≤ 1 − α.(2.2)

Re

(L(a,c)f(z))?+ 1 − α

?

≥ β

????

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

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

????

⇔

−

∞

?

n=2n(n − 1)(a)n−1

(c)n−1anzn−1+ (1 − α)

∞

?

?∞

?

n=2n(a)n−1

(c)n−1anzn−1

?

1 −

n=2n(a)n−1

(c)n−1anzn−1

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Subclass of uniformly convex functions

1843

≥ β

????????

∞

?

n=2n(n − 1)(a)n−1

∞

?

(c)n−1anzn−1

1 −

n=2n(a)n−1

(c)n−1anzn−1

????????

.

Letting z → 1 along the real axis, we obtain the desired inequality

∞

?

−1 ≤ α < 1,β ≥ 0.

Corollary 2.1 : Let the function f(z) defined by (1.7) be in the class UCT(α,β).

Then

an≤

n[n(1 + β) − (α + β)](a)n−1

Remark 2.1 : In view of Theorem 2.2, we can see that if f(z) is of the form

(1.7) and is in the class UCT(α,β) then

n=2

n[n(1 + β) − (α + β)]

(a)n−1

(c)n−1an≤ 1 − α,

(1 − α)(c)n−1

a2=

(1 − α)(c)

2(2 + β − α)(a).(2.3)

By fixing the second coefficient, we introduce a new subclass UCTb(α,β) of

UCT(α,β) and obtain the following theorems.

Let UCTb(α,β) denote the class of functions f(z) in UCT(α,β) and be of

the form

?

Theorem 2.3 Let the function f(z) defined by (2.4). Then f(z) ∈ UCTb(α,β)

if and only if

f(z) = z −

b(1 − α)(c)

2(2 + β − α)(a)z2−

∞

n=3

anzn(an≥ 0),0 ≤ b ≤ 1. (2.4)

∞

?

n=3

n[n(1 + β) − (α + β)](a)n−1

(c)n−1an≤ (1 − b)(1 − α)(2.5)

−1 ≤ α < 1,β > 0.

Proof : Substituting

a2=

b(1 − α)(c)

2(2 + β − α)(a), 0 ≤ b ≤ 1

in (2.2), we obtain

2(2 + β − α)(a)

(c)a2+

∞

?

n=3

n[n(1 + β) − (α + β)]

×(a)n−1

(c)n−1an≤ 1 − α