UNIFORMLY STARLIKE AND UNIFORMLY CONVEX FUNCTIONS PERTAINING TO SPECIAL FUNCTIONS

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Uniformly Starlike and Uniformly Convex Functions

V.B.L. Chaurasia and Amber Srivastava vol. 9, iss. 1, art. 30, 2008

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UNIFORMLY STARLIKE AND UNIFORMLY CONVEX FUNCTIONS PERTAINING TO SPECIAL

FUNCTIONS

V.B.L. CHAURASIA AMBER SRIVASTAVA

Department of Mathematics Department of Mathematics

University of Rajasthan Swami Keshvanand Institute of Technology,

Jaipur-302004, India Management and Gramothan

Jagatpura, Jaipur-302025, India EMail:amber@skit.ac.in

Received: 04 September, 2006

Accepted: 14 July, 2007

Communicated by: H.M. Srivastava 2000 AMS Sub. Class.: 30C45.

Key words: Analytic functions, Univalent functions, Starlike functions, Convex functions, Integral operator, Fox-Wright function.

Abstract: The main object of this paper is to derive the sufficient conditions for the func- tionz{pψq(z)}to be in the classes of uniformly starlike and uniformly convex functions. Similar results using integral operator are also obtained.

Acknowledgements: The authors are grateful to Professor H.M. Srivastava, University of Victoria, Canada for his kind help and valuable suggestions in the preparation of this paper.

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V.B.L. Chaurasia and Amber Srivastava

vol. 9, iss. 1, art. 30, 2008

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1. Introduction

LetAdenote the class of functions of the form

(1.1) f(z) = z+

∞

X

n=2

a_nzⁿ, that are analytic in the open unit disk∆ ={z :|z|<1}.

Also letSdenote the subclass ofAconsisting of all functionsf(z)of the form

(1.2) f(z) = z−

∞

X

n=2

a_nzⁿ, a_n≥0.

A function f ∈ A is said to be starlike of order α, 0 ≤ α < 1, if and only if Re

zf⁰(z) f(z)

> α, z ∈ ∆.Alsof of the form (1.1) is uniformly starlike, whenever _f(z)−f

(ξ) (z−ξ)f⁰(z)

≥ 0,(z,ξ) ∈ ∆×∆.This class of all uniformly starlike functions is denoted byU ST [4] (see also [5], [10] and [14]).

The functionf of the form (1.1) is uniformly convex in∆whenever Re

1 + (z−ξ)f⁰⁰(z) f⁰(z)

≥0, (z,ξ)∈∆×∆.

This class of all uniformly convex functions is denoted by U CV [3] (also refer [2], [6], [9] and [13]). Further it is said to be in the class U CV(α), α ≥ 0 if Re

1 + ^zf_f(z)⁰^(z)

≥α

zf⁰⁰(z) f⁰(z)

.

A functionf of the form (1.2) is said to be in the classU ST N(α),0≤α≤1, if Re

_f(z)−f

(ξ) (z−ξ)f⁰(z)

≥α,(z,ξ)∈∆×∆.

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In the present paper, we shall use analogues of the lemmas in [8] and [7] respec- tively in the following form.

Lemma 1.1. A functionf of the form (1.1) is in the classU ST(α), if

∞

X

n=2

[(3−α)n−2]|a_n| ≤(1−α)M₁,

whereM₁ >0is a suitable constant. In particular,f ∈U ST whenever

∞

X

n=2

(3n−2)|a_n| ≤M₁.

Lemma 1.2. A sufficient condition for a function f of the form (1.1) to be in the classU CV(α)is thatP∞

n=2n[(α+ 1)n−α]a_n≤M₂,whereM₂ >0is a suitable constant. In particular,f ∈U CV wheneverP∞

n=2n²a_n≤M₂.

The Fox-Wright function [12, p. 50, equation 1.5] appearing in the present paper is defined by

(1.3) _pψ_q(z) = _pψ_q

(a_j, α_j)_1,p; (b_j, β_j)_1,q; z

=

∞

X

n=0

Qp

j=1Γ(a_j +α_jn)zⁿ Qq

j=1Γ(b_j +β_jn)n!,

where α_j (j = 1, . . . , p) and β_j (j = 1, . . . , q) are real and positive and 1 + Pq

j=1β_j >Pp j=1α_j.

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2. Main Results

Theorem 2.1. If

q

X

j=1

|b_j|>

p

X

j=1

|a_j|+ 1, a_j >0 and 1 +

q

X

j=1

β_j >

p

X

j=1

α_j,

then a sufficient condition for the functionz{ pψq(z)} to be in the classU ST(α), 0≤α <1, is

(2.1)

3−α 1−α

pψ_q

(|a_j +α_j|, α_j)_1,p; (|b_j+β_j|, β_j)_1,q; 1

+ _pψ_q

(|a_j|, α_j)_1,p; (|b_j|, β_j)_1,q; 1

≤M1+ Qp

j=1Γa_j Qq

j=1Γb_j. Proof. Since

z{_pψ_q(z)}= Qp

j=1Γaj

Qq

j=1Γb_jz+

∞

X

n=2

Qp

j=1Γ[aj +αj(n−1)]zⁿ Qq

j=1Γ[b_j+β_j(n−1)](n−1)!

so by virtue of Lemma1.1, we need only to show that (2.2)

∞

X

n=2

[(3−α)n−2]

Qp

j=1Γ[a_j +α_j(n−1)]

Qq

j=1Γ[b_j+β_j(n−1)](n−1)!

≤(1−α)M1. Now, we have

∞

X

n=2

[(3−α)n−2]

Qp

j=1Γ[a_j+α_j(n−1)]

Qq

j=1Γ[b_j+β_j(n−1)](n−1)!

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=

∞

X

n=0

[(3−α)(n+ 2)−2]

Qp

j=1Γ[a_j+α_j(n+ 1)]

Qq

j=1Γ[b_j+β_j(n+ 1)](n+ 1)!

= (3−α)

∞

X

n=0

Qp

j=1Γ[(a_j+α_j) +nα_j] Qq

j=1Γ[(b_j+β_j) +nβ_j]n!

+ (1−α)

" _∞ X

n=0

Qp

j=1Γ(a_j+α_jn) Qq

j=1Γ(b_j +β_jn)

1 n! −

Qp j=1Γa_j Qq

j=1Γb_j

#

= (3−α) _pψ_q

(|a_j +α_j|, α_j)_1,p; (|b_j +β_j|, β_j)_1,q; 1

+ (1−α) _pψ_q

(|a_j|, α_j)_1,p; (|b_j|, β_j)_1,q; 1

−(1−α) Qp

j=1Γaj

Qq j=1Γbj

≤(1−α)M₁

which in view of Lemma1.1gives the desired result.

Theorem 2.2. If

q

X

j=1

b_j >

p

X

j=1

a_j+ 1, a_j >0 and 1 +

q

X

j=1

β_j >

p

X

j=1

α_j,

then a sufficient condition for the functionz{_pψ_q(z)}to be in the classU ST N(α), 0≤α <1, is:

3−α 1−α

pψ_q

(a_j +α_j, α_j)_1,p; (b_j+β_j, β_j)_1,q; 1

+ _pψ_q

(a_j, α_j)_1,p; (b_j, β_j)_1,q; 1

≤M₁+ Qp

j=1Γa_j Qq

j=1Γb_j. Proof. The proof of Theorem2.2is a direct consequence of Theorem2.1.

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Theorem 2.3. If

q

X

j=1

bj >

p

X

j=1

aj + 2, aj >0 and 1 +

q

X

j=1

βj >

p

X

j=1

αj,

then a sufficient condition for the function z{_pψ_q(z)} to be in the classU CV(α), 0≤α <1, is

(2.3) (1 +α)pψq

(a_j+ 2α_j, α_j)_1,p; (b_j + 2β_j, β_j)_1,q; 1

+ (2α+ 3) _pψ_q

(a_j+α_j, α_j)_1,p; (b_j +β_j, β_j)_1,q; 1

+ _pψ_q(1)≤M₂+ Qp

j=1Γaj

Qq

j=1Γb_j. Proof. By virtue of Lemma1.2, it suffices to prove that

(2.4)

∞

X

n=2

n[(α+ 1)n−α]

Qp

j=1Γ[aj +αj(n−1)]

Qq

j=1Γ[b_j+β_j(n−1)](n−1)! ≤M₂. Now, we have

(2.5)

∞

X

n=2

n[(α+ 1)n−α]

Qp

j=1Γ[aj +αj(n−1)]

Qq

j=1Γ[b_j +β_j(n−1)](n−1)!

= (1 +α)

∞

X

n=1

(n+ 1)² Qp

j=1Γ(a_j +α_jn) Qq

j=1Γ[(b_j+β_jn)n!

−α

∞

X

n=1

(n+ 1) Qp

j=1Γ(a_j+α_jn) Qq

j=1Γ(b_j+β_jn)n!.

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Using(n+ 1)² =n(n+ 1) + (n+ 1), (2.5) may be expressed as (1+α)

∞

X

n=1

(n+ 1)

Qp

j=1Γ(b_j +β_jn)(n−1)!

(2.6)

+

∞

X

n=1

(n+ 1) Qp

j=1Γ(b_j +β_jn)n!

= (1 +α)

∞

X

n=2

Qp

j=1Γ(aj +αjn) Qq

j=1Γ(b_j +β_jn)(n−2)!

+ (2α+ 3)

∞

X

n=0

Qp

j=1Γ[(a_j +α_j) +α_jn]

Qq

j=1Γ[(b_j+β_j) +β_jn]n!

+

∞

X

n=1

Qp

j=1Γ(a_j+α_jn) Qq

j=1Γ(b_j+β_jn)n!

= (1 +α) _pψ_q

(a_j + 2α_j, α_j)_1,p; (b_j+ 2β_j, β_j)_1,q; 1

+ (2α+ 3) _pψ_q

(a_j+α_j, α_j)_1,p; (bj +βj, βj)1,q; 1

+ _pψ_q(1)− Qp

j=1Γa_j Qq

j=1Γb_j, which is bounded above by M₂ if and only if (2.3) holds. Hence the theorem is proved.

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3. An Integral Operator

In this section we obtain sufficient conditions for the function

pφ_q

(a_j, α_j)_1,p; (bj, βj)1,q; z

= Z z

0

pψ_q(x)dx to be in the classesU ST andU CV.

Theorem 3.1. If

q

X

j=1

b_j >

p

X

j=1

a_j, a_j >0 and 1 +

q

X

j=1

β_j >

p

X

j=1

α_j,

then a sufficient condition for the functionpφ_q(z) = Rz

0 pψ_q(x)dxto be in the class U ST is

(3.1) 3 pψq(1)−2pψq

(a_j−α_j, α_j)_1,p; (b_j −β_j, β_j)_1,q; 1

+ 2 Qp

j=1Γ(a_j −α_j) Qq

j=1Γ(b_j −β_j) ≤M₁+ Qp

j=1Γa_j Qq

j=1Γb_j. Proof. Since

pφ_q(z) = Z z

0

pψ_q(x)dx (3.2)

= Qp

j=1Γa_j Qq

j=1Γb_jz+

∞

X

n=2

Qp

j=1Γ[(a_j −α_j) +α_jn]

Qq

j=1Γ[(b_j−β_j) +β_jn]

zⁿ n!,

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we have

∞

X

n=2

(3n−2) Qp

j=1Γ[(a_j −α_j) +α_jn]

Qq

j=1Γ[(b_j −β_j) +β_jn]n!

(3.3)

= 3

∞

X

n=1

Qp

j=1Γ(aj+αjn) Qq

j=1Γ(b_j+β_jn)n! −2

" _∞ X

n=0

Qp

j=1Γ[(aj −αj) +αjn]

Qq

j=1Γ[(b_j−β_j) +β_jn]n!

− Qp

j=1Γ(a_j−α_j) Qq

j=1Γ(b_j−β_j) − Qp

j=1Γa_j Qq

j=1Γb_j

#

= 3 _pψ_q(1)−2 _pψ_q

(aj −αj, αj)1,p; (b_j−β_j, β_j)_1,q; 1

+ 2 Qp

j=1Γ(a_j −α_j) Qq

j=1Γ(b_j −β_j) − Qp

j=1Γa_j Qq

j=1Γb_j. In view of Lemma1.1, (3.3) leads to the result (3.1).

Theorem 3.2. If

q

X

j=1

b_j >

p

X

j=1

a_j, a_j >0 and 1 +

q

X

j=1

β_j >

p

X

j=1

α_j,

then a sufficient condition for the functionpφ_q(z) = Rz

0 pψ_q(x)dxto be in the class U CV is

(3.4) pψ_q

(a_j +α_j, α_j)_1,p; (bj+βj, βj)1,q; 1

+ _pψ_q(1) ≤M₂+ Qp

j=1Γa_j Qq

j=1Γb_j.

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Proof. Sincepφ_q(z)has the form (3.2), then

∞

X

n=2

n² Qp

j=1Γ[(aj −αj) +αjn]

Qq

j=1Γ[(bj−βj) +βjn]n!

(3.5)

=

∞

X

n=1

(n+ 1) Qp

j=1Γ(b_j +β_jn)n!

=

∞

X

n=0

Qp

j=1Γ[(a_j +α_j) +α_jn]

Qq

j=1Γ[(b_j+β_j) +β_jn]n! +

∞

X

n=0

Qp

j=1Γ(b_j +β_jn)n!− Qp

j=1Γa_j Qq

j=1Γb_j

= pψq

(a_j+α_j, α_j)_1,p; (b_j +β_j, β_j)_1,q; 1

+ pψq(1)− Qp

j=1Γa_j Qq

j=1Γb_j, which in view of Lemma1.2gives the desired result (3.4).

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4. Particular Cases

4.1. By settingα₁ =α₂ =· · ·=α_p = 1;β₁ =β₂ =· · ·=β_q= 1and M₁ =M₂ =M₃ =

Qp j=1Γa_j Qq

j=1Γb_j,

Theorems 2.1, 2.3, 3.1 and 3.2 reduce to the results recently obtained by Shan- mugam, Ramachandran, Sivasubramanian and Gangadharan [11].

4.2. By specifying the parameters suitably, the results of this paper readily yield the results due to Dixit and Verma [1].

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References

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[2] A. GANGADHARAN, T.N. SHANMUGAMANDH.M. SRIVASTAVA, Gen- eralized hypergeometric functions associated with k-uniformly convex func- tions, Comput. Math. Appl., 44 (2002), 1515–1526.

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[9] C. RAMACHANDRAN, T.N. SHANMUGAM, H.M. SRIVASTAVAAND A.

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