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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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≥ 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

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S. M. Khairnar and N. H. More

which gives

∞

?

n=3

n[n(1 + β) − (α + β)](a)n−1

(c)n−1an≤ (1 − b)(1 − α)

which is the desired result.

Corollary 2.2 : Let the function f(z) defined by (2.4) be in the class UCTb(α,β).

Then

an≤

n ≥ 3,−1 ≤ α < 1,β ≥ 0.

Theorem 2.4 : The class UCTb(α,β) is closed under convex linear combina-

tion.

Proof : Let the function f(z) be defined by (2.4) and g(z) defined by

(1 − b)(1 − α)(c)n−1

n[n(1 + β) − (α + β)](a)n−1, (2.6)

g(z) = z −

b(1 − α)(c)

2(2 + b − α)(a)z2−

∞

?

n=3

dnzn, (2.7)

where dn≥ 0 and 0 ≤ b ≤ 1.

Assuming that f(z) and g(z) are in the class UCTb(α,β), it is sufficient to

prove that the function H(z) defined by

H(z = λf(z) + (1 − λ)g(z), 0 ≤ λ ≤ 1

is also in the class UCTb(α,β).

Since

(2.8)

H(z) = z −

b(1 − α)(c)

2(2 + β − α)(a)z2

∞

?

−

n=3

{λn+ (1 − λ)dn}zn,(2.9)

an≥ 0,dn≥ 0,0 ≤ b ≤ 1, we observe that

∞

?

≤ (1 − b)(1 − α)

which is, in view of Theorem 2.3, implies that H(z) ∈ UCTb(α,β).

This completes the proof of the theorem.

Theorem 2.5 : Let the functions

n=3

n[n(1 + β) − (α + β)](a)n−1

(c)n−1{λan+ (1 − λ)dn}

(2.10)

fj(z) = z −

b(1 − α)(c)

2(2 + β − α)(a)z2−

∞

?

n=3

an,jzn, (2.11)

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Subclass of uniformly convex functions

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an,j≥ 0 be in the class UCTb(α,β) for every j (j = 1,2,3,···,m). Then the

function F(z) defined by

F(z) =

m

?

j=1

μjfj(z), (2.12)

is also in the class UCTb(α,β), where

∞

?

j=1

μj= 1. (2.13)

Proof : Combining the definitions (2.11) and (2.12) further by (2.13) we have

F(z) = z −

b(1 − α)(c)

2(2 + β − α)(a)≥ 2 −

∞

?

n=3

?

m

?

j=1

μjan,j

?

zn. (2.14)

Since fj(z) ∈ UCTn(α,β) for every j = 1,2,···,m, Theorem 2.3 yields

∞

?

Thus we obtain

n=3

n[n(1 + β) − (α + β)](a)n−1

(c)n−1an,j≤ (1 − b)(1 − α). (2.15)

∞

?

n=3

n[n(1 + β) − (α + β)](a)n−1

?∞

n=3

≤ (1 − b)(1 − α)

(c)n−1

?

m

?

?

j=1

μjan,j

?

=

m

?

j=1

?

n[n(1 + β) − (α + β)]

(a)n−1

(c)n−1an,j

in view of Theorem 2.3. So, F(z) ∈ UCTb(α,β).

Theorem 2.6 : Let

f2(z) = z −

b(1 − α)(c)

2(2 + β − α)(a)z2

(2.16)

and

fn(z) = z −

b(1 − α)(c)

2(2 + β − α)(a)z2−

(1 − b(1 − α)(c)n−1

n[n(1 + β) − (α + β)(a)n−1zn

(2.17)

for n = 3,4,···. Then f(z) is in the class UCTb(α,β) if and only if it can be

expressed in the form

∞

?

f(z) =

n=2

λnfn(z),(2.18)

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S. M. Khairnar and N. H. More

where λn≥ 0 and

Proof : we suppose that f(z) can be expressed in the form (2.18). Then we

have

∞

?

n=2λn= 1.

f(z) = z −

b(1 − α)(c)

2(2 + β − α)(a)z2

∞

?

= z −

n=2

−

n=3

λn

(1 − b)(1 − α)(c)n−1

n[n(1 + β) − (α + β)](a)n−1zn

∞

?

Anzn, (2.19)

where

A2=

b(1 − α)(c)

2(2 + β − α)

(2.20)

An=

λn(1 − b)(1 − α)(c)n−1

n[n(1 + β) − (α + β)](a)n−1, n = 3,4,···. (2.21)

Therefore,

∞

?

= b(1 − α)

n=2

n[n(1 + β) − (α + β)](a)n−1

(c)n−1An

∞

?

n=3

λn(1 − b)(1 − α)

= (1 − α)[b + (1 − λ2)(1 − b)]

≤ (1 − α), (2.22)

It follows from Theorem 2.2 and Theorem 2.3 that f(z) is in the class UCTb(α,β).

Conversely, we suppose that f(z) defined by (2.4) is in the class UCTb(α,β).

Then by using (2.6), we get

an≤

(1 − b)(1 − α)(c)n−1

n[n(1 + β) − (α + β)](a)n−1, (n ≥ 3). (2.23)

Setting

λn=n[n(1 + β) − (α + β)](a)n−1

(1 − b)(1 − α)(c)n−1

, (n ≥ 3) (2.24)

and

λ2= 1 −

∞

?

n=3

λn,

we have (2.18). This completes the proof of Theorem 2.6.

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Corolalry 2.3 : The extreme points of the class UCTb(α,β) are functions

fn(z),n ≥ 2 given by Theorem 2.6.

3. The Class UCTbn,k(α,β)

Instead of fixing just the second coefficient, we can fix finitely many coef-

ficients. Let UCTbn,k(α,β) be the class of functions of the form

f(z) = z −

k

?

n=2

bn(1 − α)(c)n−1

n[n(1 + β) − (α + β)](a)n−1zn−

∞

?

n=k+1

anzn; (3.1)

where 0 ≤

Theorem 3.1 : The extreme points of the class UCTbn,k(α,β) are

k ?

n=2bn= b ≤ 1. Note that UCTb2,2(α,β) = UCTb(α,β).

fk(z) = z −

k

?

n=2

bn(1 − α)(c)n−1

n[n(1 + β) − (α + β)](a)n−1zn

and

f(n(z) = z −

∞

?

∞

?

∞

?

∞

?

n=2

bn(1 − α)(c)n−1

n[n(1 + β) − (α + β)](a)n−1zn

(1 − b)(1 − α)(c)n−1

n[n(1 + β) − (α + β)](a)n−1zn.

bn(1 − α)(c)n−1

n[n(1 + β) − (α + β)](an−1zn

(1 − b)(1 − α)(c)n−1

n[n(1 + β) − (α + β)](a)n−1zn.

−

n=k+1

= z −

n=2

−

n=k+1

The details of the proof are omitted, since the characterization of the ex-

treme points enables us to solve the standard extremal problems in the same

manner as was done for UCTb(α,β).

References

[1] B. C. Carlson, and D. B. Shaffer, Starlike and prestarlike hyper-

geometric functions, SIAM J. Math., Anal., 15 (1984), 737-745.

[2] A.W. Goodman, on uniformly convex functions, Ann. Polon

Math., 56 (1991), 87-92.

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S. M. Khairnar and N. H. More

[3] A. W. Goodman, On uniformly starlike functions, J. Math.

Anal. & Appl., 155 (1991), 364-370.

[4] S.M.Khairnar and Meena More, On certain subclass of analytic

functions involving Al-Oboudi differential operator, J. Inequal.

Pure and Appl. Math., 10(1), Art. 57,(2009), 11pp.

[5] S.M.Khairnar and Meena More, Some applications and proper-

ties of Generalized fractional calculus Operators to a subclass

of analytic and Multivalent functions, Korean Journal of Math-

ematics,17(2), (2009), 127-145.

[6] S.M.Khairnar and Meena More, Properties of certain classes of

analytic functions defined by Srivastava-Attiya operator, Demon-

stratio Mathematica, (Accepted to appear in 2010).

[7] St. Ruscheweyh, St. and V. Singh, On the order of starlikeness

of hypergeometric functions, J. Math. Anal. Appl., (1986),

113.

[8] F. Rønning, Uniformly convex functions and a corresponding

class of starlike functions, Proc. Amer. Math. Soc., 118 (1993),

189-196.

[9] S. Ruscheweyh, New criteria for univalent funtions, Proc. Amer.

Math. Soc., 49 (1975), 109-115.

Received: March, 2009