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volume 6, issue 3, article 71, 2005.

Received 12 May, 2005;

accepted 24 May, 2005.

Communicated by:S. Saitoh

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Journal of Inequalities in Pure and Applied Mathematics

ON THE FEKETE-SZEGÖ PROBLEM FOR SOME SUBCLASSES OF ANALYTIC FUNCTIONS

T.N. SHANMUGAM AND S. SIVASUBRAMANIAN

Department of Mathematics College of Engineering

Anna University, Chennai-600 025 Tamilnadu, India.

EMail:[email protected] Department of Mathematics Easwari Engineering College Ramapuram, Chennai-600 089 Tamilnadu, India.

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 165-05

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On the Fekete-Szegö Problem for Some Subclasses of

Analytic Functions

T.N. Shanmugam and S. Sivasubramanian

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J. Ineq. Pure and Appl. Math. 6(3) Art. 71, 2005

http://jipam.vu.edu.au

Abstract

In this present investigation, the authors obtain Fekete-Szegö’s inequality for certain normalized analytic functions f(z) defined on the open unit disk for which (1−α)f(z)+αzf^zf⁰^(z)+αz²^f⁰⁰^(z)⁰(z) (α ≥ 0)lies in a region starlike with respect to1and is symmetric with respect to the real axis. Also certain applications of the main result for a class of functions defined by convolution are given. As a special case of this result, Fekete-Szegö’s inequality for a class of functions defined through fractional derivatives is obtained. The Motivation of this paper is to give a generalization of the Fekete-Szegö inequalities obtained by Srivastava and Mishra .

2000 Mathematics Subject Classification:Primary 30C45

Key words: Analytic functions, Starlike functions, Subordination, Coefficient prob- lem, Fekete-Szegö inequality.

1. Introduction

LetAdenote the class of all analytic functionsf(z)of the form (1.1) f(z) =z+

∞

X

k=2

a_kz^k (z ∈∆ :={z ∈C| |z|<1})

and S be the subclass of A consisting of univalent functions. Let φ(z) be an analytic function with positive real part on∆withφ(0) = 1, φ⁰(0) > 0which maps the unit disk∆onto a region starlike with respect to1which is symmetric with respect to the real axis. LetS^∗(φ)be the class of functions inf ∈ S for which

zf⁰(z)

f(z) ≺φ(z), (z ∈∆) andC(φ)be the class of functions inf ∈ S for which

1 + zf⁰⁰(z)

f⁰(z) ≺φ(z), (z ∈∆),

where ≺denotes the subordination between analytic functions. These classes were introduced and studied by Ma and Minda [10]. They have obtained the Fekete-Szegö inequality for the functions in the classC(φ). Sincef ∈ C(φ)if and only if zf⁰(z) ∈ S^∗(φ), we get the Fekete-Szegö inequality for functions in the class S^∗(φ). For a brief history of the Fekete-Szegö problem for the class of starlike, convex and close-to-convex functions, see the recent paper by Srivastava et al. [7].

In the present paper, we obtain the Fekete-Szegö inequality for functions in a more general classMα(φ)of functions which we define below. Also we give

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applications of our results to certain functions defined through convolution (or the Hadamard product) and in particular we consider a class M_α^λ(φ) of functions defined by fractional derivatives. The motivation of this paper is to give a generalization of the Fekete-Szegö inequalities of Srivastava and Mishra [6].

Definition 1.1. Letφ(z)be a univalent starlike function with respect to 1 which maps the unit disk ∆onto a region in the right half plane which is symmetric with respect to the real axis,φ(0) = 1andφ⁰(0) > 0. A functionf ∈ Ais in the classMα(φ)if

zf⁰(z) +αz²f⁰⁰(z)

(1−α)f(z) +αzf⁰(z) ≺φ(z) (α≥0).

For fixedg ∈ A, we define the classM_α^g(φ)to be the class of functionsf ∈ A for which(f∗g)∈M_α(φ).

To prove our main result, we need the following:

Lemma 1.1. [10] Ifp₁(z) = 1 +c₁z+c₂z²+· · · is an analytic function with positive real part in∆, then

|c2−vc²₁| ≤











−4v+ 2 ifv ≤0;

2 if0≤v ≤1;

4v−2 ifv ≥1.

Whenv <0orv >1, the equality holds if and only ifp₁(z)is(1 +z)/(1−z) or one of its rotations. If0< v <1, then the equality holds if and only ifp₁(z)

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is(1 +z²)/(1−z²)or one of its rotations. Ifv = 0, the equality holds if and only if

p1(z) = 1

2 +1 2λ

1 +z 1−z +

1 2 −1

2λ

1−z

1 +z (0≤λ≤1)

or one of its rotations. If v = 1, the equality holds if and only if p₁ is the reciprocal of one of the functions such that the equality holds in the case of v = 0.

Also the above upper bound is sharp, and it can be improved as follows when 0< v < 1:

|c₂−vc²₁|+v|c₁|² ≤2

0< v ≤ 1 2

and

|c2−vc²₁|+ (1−v)|c1|² ≤2 1

2 < v ≤1

.

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2. Fekete-Szegö Problem

Our main result is the following:

Theorem 2.1. Letφ(z) = 1 +B₁z+B₂z²+B₃z³+· · ·. Iff(z)given by (1.1) belongs toMα(φ), then

|a₃−µa²₂| ≤











B2

2(1+2α) − _(1+α)^µ 2B₁²+ 2(1+2α)(1+α)¹ ²B₁² if µ≤σ₁;

B1

2(1+2α) if σ₁ ≤µ≤σ₂;

−_2(1+2α)^B² +_(1+α)^µ 2B₁²− 2(1+2α)(1+α)¹ ²B₁² if µ≥σ₂, where

σ₁ := (1 +α)²(B2−B1) + (1 +α²)B₁² 2(1 + 2α)B₁² , σ₂ := (1 +α)²(B₂+B₁) + (1 +α²)B₁²

2(1 + 2α)B₁² . The result is sharp.

Proof. Forf(z)∈M_α(φ), let

(2.1) p(z) := zf⁰(z) +αz²f⁰⁰(z)

(1−α)f(z) +αzf⁰(z) = 1 +b₁z+b₂z²+· · ·. From (2.1), we obtain

(1 +α)a2 =b1 and (2 + 4α)a3 =b2+ (1 +α²)a²₂.

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Sinceφ(z)is univalent andp≺φ, the function p₁(z) = 1 +φ⁻¹(p(z))

1−φ⁻¹(p(z)) = 1 +c₁z+c₂z²+· · · is analytic and has a positive real part in∆. Also we have

(2.2) p(z) =φ

p1(z)−1 p₁(z) + 1

and from this equation (2.2), we obtain b₁ = 1

2B₁c₁ and

b₂ = 1

2B₁(c₂− 1

2c²₁) + 1 4B₂c²₁. Therefore we have

(2.3) a₃ −µa²₂ = B₁

4(1 + 2α)

c₂ −vc²₁ , where

v := 1 2

1−B₂

B₁ +(2µ−1) +α(4µ−α) (1 +α)² B1

.

Our result now follows by an application of Lemma 1.1. To show that the bounds are sharp, we define the functionsK_α^φⁿ (n= 2,3, . . .)by

z[K_α^φⁿ]⁰(z) +αz²[K_α^φⁿ]⁰⁰(z)

(1−α)[Kα^φⁿ](z) +αz[Kα^φⁿ]⁰(z) =φ(zⁿ⁻¹), K_α^φⁿ(0) = 0 = [K_α^φⁿ]⁰(0)−1

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and the functionF_α^λ andG^λ_α(0≤λ≤1)by z[F_α^λ]⁰(z) +αz²[F_α^λ]⁰⁰(z)

(1−α)[F_α^λ](z) +αz[F_α^λ]⁰(z) =φ

z(z+λ) 1 +λz

, F^λ(0) = 0 = (F^λ)⁰(0)−1 and

z[G^λ_α]⁰(z) +αz²[G^λ_α]⁰⁰(z)

(1−α)[G^λ_α](z) +αz[G^λ_α]⁰(z) =φ

−z(z+λ) 1 +λz

, G^λ(0) = 0 = (G^λ)⁰(0).

Clearly the functionsK_α^φn, F_α^λ, G^λ_α ∈M_α(φ). Also we writeK_α^φ:=K_α^φ². Ifµ < σ₁ orµ > σ₂, then the equality holds if and only iff isK_α^φor one of its rotations. When σ₁ < µ < σ₂, the equality holds if and only iff isK_α^φ³ or one of its rotations. If µ= σ₁ then the equality holds if and only if f isF_α^λ or one of its rotations. If µ= σ₂ then the equality holds if and only iff isG^λ_αor one of its rotations.

Remark 1. If σ₁ ≤ µ ≤ σ₂, then, in view of Lemma 1.1, Theorem2.1 can be improved. Letσ₃be given by

σ₃ := (1 +α)²B2+ (1 +α²)B₁² 2(1 + 2α)B₁² . Ifσ₁ ≤µ≤σ₃, then

|a₃−µa²₂|+ (1 +α)² 2(1 + 2α)B₁²

B₁−B₂+ (2µ−1) +α(4µ−α) (1 + 2α) B₁²

|a₂|²

≤ B1

2(1 + 2α).

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Ifσ₃ ≤µ≤σ₂, then

|a₃−µa²₂|+ (1 +α)² 2(1 + 2α)B₁²

B₁+B₂− (2µ−1) +α(4µ−α) (1 + 2α)² B₁²

|a₂|²

≤ B₁ 2(1 + 2α).

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3. Applications to Functions Defined by Fractional Derivatives

In order to introduce the classM_α^λ(φ), we need the following:

Definition 3.1 (see [3, 4]; see also [8, 9]). Let f(z) be analytic in a simply connected region of thez-plane containing the origin. The fractional derivative off of orderλis defined by

D_z^λf(z) := 1 Γ(1−λ)

d dz

Z z 0

f(ζ)

(z−ζ)^λdζ (0≤λ <1),

where the multiplicity of(z−ζ)^λis removed by requiring thatlog(z−ζ)is real forz−ζ >0.

Using the above Definition3.1and its known extensions involving fractional derivatives and fractional integrals, Owa and Srivastava [3] introduced the op- eratorΩ^λ :A → Adefined by

(Ω^λf)(z) = Γ(2−λ)z^λD^λ_zf(z), (λ 6= 2,3,4, . . .).

The class M_α^λ(φ) consists of functions f ∈ A for which Ω^λf ∈ M_α(φ).

Note that M₀⁰(φ) ≡ S^∗(φ) andM_α^λ(φ)is the special case of the class M_α^g(φ) when

(3.1) g(z) =z+

∞

X

n=2

Γ(n+ 1)Γ(2−λ) Γ(n+ 1−λ) zⁿ.

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Let

g(z) = z+

∞

X

n=2

g_nzⁿ (g_n>0).

Since

f(z) =z+

∞

X

n=2

a_nzⁿ ∈M_α^g(φ) if and only if

(f∗g) = z+

∞

X

n=2

g_na_nzⁿ∈M_α(φ),

we obtain the coefficient estimate for functions in the class M_α^g(φ), from the corresponding estimate for functions in the classM_α(φ). Applying Theorem2.1 for the function(f ∗g)(z) =z+g2a2z²+g3a3z³+· · ·, we get the following Theorem3.1after an obvious change of the parameterµ:

Theorem 3.1. Let the function φ(z) be given by φ(z) = 1 + B₁z +B₂z² + B3z³+· · ·. Iff(z)given by (1.1) belongs toM_α^g(φ), then

|a₃−µa²₂|

≤











1 g3

h B2

2(1+2α)− _(1+α)^µg³2g²₂B²₁+ 2(1+2α)(1+α)¹ ²B²₁i

if µ≤σ₁;

1 g3

h B1

2(1+2α)

i

if σ₁ ≤µ≤σ₂;

1 g3

h−_2(1+2α)^B² + _(1+α)^µg³2g₂²B₁²− 2(1+2α)(1+α)¹ ²B₁²i

if µ≥σ₂,

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where

σ₁ := g²₂ g₃

(1 +α)²(B₂−B₁) + (1 +α²)B₁² 2(1 + 2α)B₁²

σ₂ := g²₂ g₃

(1 +α)²(B₂+B₁) + (1 +α²)B₁² 2(1 + 2α)B₁² . The result is sharp.

Since

(Ω^λf)(z) =z+

∞

X

n=2

Γ(n+ 1)Γ(2−λ) Γ(n+ 1−λ) a_nzⁿ, we have

(3.2) g2 := Γ(3)Γ(2−λ)

Γ(3−λ) = 2 2−λ and

(3.3) g₃ := Γ(4)Γ(2−λ)

Γ(4−λ) = 6

(2−λ)(3−λ).

Forg₂ andg₃ given by (3.2) and (3.3), Theorem3.1reduces to the following:

Theorem 3.2. Let the function φ(z) be given by φ(z) = 1 + B1z +B2z² + B₃z³+· · ·. Iff(z)given by (1.1) belongs toM_α^λ(φ), then

|a3−µa²₂| ≤











(2−λ)(3−λ)

6 γ if µ≤σ1;

(2−λ)(3−λ)

6 · _2(1+2α)^B¹ if σ1 ≤µ≤σ2;

(2−λ)(3−λ)

6 γ if µ≥σ2,

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where

γ := B₂

2(1 + 2α) − 3(2−λ) 2(3−λ)

µ

(1 +α)²B₁²+ 1

2(1 + 2α)(1 +α)²B₁²,

σ1 := 2(3−λ)

3(2−λ).(1 +α)²(B₂−B₁) + (1 +α²)B₁² 2(1 + 2α)B₁²

σ₂ := 2(3−λ)

3(2−λ).(1 +α)²(B2+B1) + (1 +α²)B₁² 2(1 + 2α)B₁² . The result is sharp.

Remark 2. When α = 0, B₁ = 8/π² and B₂ = 16/(3π²), the above Theo- rem 3.1 reduces to a recent result of Srivastava and Mishra[6, Theorem 8, p.

64] for a class of functions for which Ω^λf(z) is a parabolic starlike function [2,5].

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References

[1] B.C. CARLSON AND D.B. SHAFFER, Starlike and prestarlike hyperge- ometric functions, SIAM J. Math. Anal., 15 (1984), 737–745.

[2] A.W. GOODMAN, Uniformly convex functions, Ann. Polon. Math., 56 (1991), 87–92.

[3] S. OWA AND H.M. SRIVASTAVA, Univalent and starlike generalized bypergeometric functions, Canad. J. Math., 39 (1987), 1057–1077.

[4] S. OWA, On the distortion theorems I, Kyungpook Math. J., 18 (1978), 53–58.

[5] F. RØNNING, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), 189–196.

[6] H.M. SRIVASTAVA AND A.K. MISHRA, Applications of fractional cal- culus to parabolic starlike and uniformly convex functions, Computer Math. Appl., 39 (2000), 57–69.

[7] H.M. SRIVASTAVA, A.K. MISHRA ANDM.K. DAS, The Fekete-Szegö problem for a subclass of close-to-convex functions, Complex Variables, Theory Appl., 44 (2001), 145–163.

[8] H.M. SRIVASTAVAANDS. OWA, An application of the fractional deriva- tive, Math. Japon., 29 (1984), 383–389.

[9] H.M. SRIVASTAVA AND S. OWA, Univalent functions, Fractional Cal- culus, and their Applications, Halsted Press/John Wiley and Songs, Chich- ester/New York, (1989).

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[10] W. MA AND D. MINDA, A unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analy- sis, Z. Li, F. Ren, L. Yang, and S. Zhang(Eds.), Int. Press (1994), 157–169.