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volume 7, issue 4, article 145, 2006.

Received 06 February, 2006;

accepted 25 August, 2006.

Communicated by:G. Kohr

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Journal of Inequalities in Pure and Applied Mathematics

A COEFFICIENT INEQUALITY FOR CERTAIN CLASSES OF ANALYTIC FUNCTIONS OF COMPLEX ORDER

K. SUCHITHRA, B. ADOLF STEPHEN AND S. SIVASUBRAMANIAN

Department of Applied Mathematics Sri Venkateswara College of Engineering Sriperumbudur, Chennai - 602105, India.

EMail:suchithrak@svce.ac.in Department of Mathematics Madras Christian College

Tambaram, Chennai - 600059, India.

EMail:adolfmcc2003@yahoo.co.in Department of Mathematics Easwari Engineering College Ramapuram, Chennai - 600089, India.

EMail:sivasaisastha@rediffmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 032-06

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A Coefficient Inequality For Certain Classes Of Analytic Functions Of Complex Order

K. Suchithra, B. Adolf Stephen and S. Sivasubramanian

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J. Ineq. Pure and Appl. Math. 7(4) Art. 145, 2006

Abstract

In the present investigation, we obtain the Fekete-Szegö inequality for a certain normalized analytic functionf(z)defined on the open unit disk for which1 +

1 b

hzf⁰(z)+αz²f⁰⁰(z)

f(z) −1

i

(α ≥ 0andb6= 0, a complex number) lies in a region starlike with respect to 1 and symmetric with respect to real axis. Also certain application of the main result for a class of functions of complex order defined by convolution is given. The motivation of this paper is to give a generalization of the Fekete-Szegö inequalities for subclasses of starlike functions of complex order.

2000 Mathematics Subject Classification:Primary 30C45.

Key words: Starlike functions of complex order, Convex functions of complex order, Subordination, Fekete-Szegö inequality.

1. Introduction

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

∞

X

k=2

akz^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 to 1 which 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 functionsf ∈ Sfor 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 [4]. They have obtained the Fekete-Szegö inequality for functions in the class C(φ). Since f ∈ C(φ) iff zf⁰(z) ∈ S^∗(φ), we get the Fekete-Szegö inequality for functions in the class S^∗(φ).

The classS_b^∗(φ)consists of all analytic functionsf ∈ Asatisfying 1 + 1

b

zf⁰(z) f(z) −1

≺φ(z)

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and the classC_b(φ)consists of functionsf ∈ Asatisfying 1 + 1

b

zf⁰⁰(z) f⁰(z)

≺φ(z).

These classes were defined and studied by Ravichandran et al. [7]. They have obtained the Fekete-Szegö inequalities for functions in these classes.

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.

[10].

In the present paper, we obtain the Fekete-Szegö inequality for functions in a more general classM_α,b(φ)of functions which we define below. Also we give applications of our results to certain functions defined through convolution (or Hadamard product) and in particular we consider a classM_α,b^λ (φ)of functions defined by fractional derivatives.

Definition 1.1. Letb6= 0be a complex number. Letφ(z)be an analytic function with positive real part on∆withφ(0) = 1,φ⁰(0)>0 which maps the unit disk

∆onto a region starlike with respect to 1 which is symmetric with respect to the real axis. A functionf ∈ Ais in the classM_α,b(φ)if

1 + 1 b

zf⁰(z) +αz²f⁰⁰(z) f(z) −1

≺φ(z) (α≥0).

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

To prove our result, we need the following:

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Lemma 1.1 ([7]). Ifp(z) = 1 +c₁z+c₂z²+· · · is a function with positive real part, then for any complex numberµ,

|c₂−µc²₁| ≤2 max{1,|2µ−1|}

and the result is sharp for the functions given by

p(z) = 1 +z²

1−z², p(z) = 1 +z 1−z.

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2. The 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 toM_α,b(φ), then

|a₃−µa²₂| ≤ B1|b|

2(1 + 3α)max

1,

B2

B₁ +

(1 + 2α)−2µ(1 + 3α) (1 + 2α)²

bB₁

. The result is sharp.

Proof. Iff(z)∈M_α,b(φ), then there is a Schwarz functionw(z), analytic in∆ withw(0) = 0and|w(z)|<1in∆such that

(2.1) 1 + 1

b

zf⁰(z) +αz²f⁰⁰(z)

f(z) −1

=φ(w(z)).

Define the functionp₁(z)by

(2.2) p₁(z) := 1 +w(z)

1−w(z) = 1 +c₁z+c₂z²+· · · .

Since w(z) is a Schwarz function, we see that Rp₁(z) > 0 and p₁(0) = 1.

Define the functionp(z)by (2.3) p(z) := 1 +1

b

zf⁰(z) +αz²f⁰⁰(z) f(z) −1

= 1 +b₁z+b₂z²+· · · .

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In view of the equations (2.1), (2.2), (2.3), we have

(2.4) p(z) =φ

p₁(z)−1 p1(z) + 1

and from this equation (2.4), we obtain

(2.5) b₁ = 1

2B₁c₁ and

(2.6) b₂ = 1

2B₁

c₂−1 2c²₁

+1

4B₂c²₁. From equation (2.3), we obtain

(1 + 2α)a2 =bb1,

(2 + 6α)a₃ =bb₂+ (1 + 2α)a²₂ or equivalently we have

(2.7) a₂ = bb₁

1 + 2α,

(2.8) a3 = 1

2 + 6α

bb2+ b²b²₁ 1 + 2α

.

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Applying (2.5) in (2.7) and (2.5), (2.6) in (2.8), we have a₂ = bB1c1

2(1 + 2α), a₃ = bB₁c₂

4(1 + 3α)+ c²₁ 8(1 + 3α)

b²B₁²

1 + 2α −b(B₁−B₂)

. Therefore we have

(2.9) a₃ −µa²₂ = bB1

4(1 + 3α)

c₂ −vc²₁ , where

v := 1 2

1− B2

B₁ +

2µ(1 + 3α)−(1 + 2α) (1 + 2α)²

bB₁

.

Our result now follows by an application of Lemma1.1. The result is sharp for the function defined by

1 + 1 b

zf⁰(z) +αz²f⁰⁰(z) f(z) −1

=φ(z²) and

1 + 1 b

zf⁰(z) +αz²f⁰⁰(z) f(z) −1

=φ(z).

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Example 2.1. By taking b = (1− β)e^−iλcosλ, φ(z) = ^1+z_1−z, we obtain the following sharp inequality

|a₃−µa²₂| ≤ (1−β) cosλ 1 + 3α

×max

1,

e^iλ−2

2µ(1 + 3α)−(1 + 2α) (1 + 2α)²

(1−β) cosλ

. Remark 1. Whenα= 0, Example2.1reduces to a result of [7] forλ-spirallike functionf(z)of orderβ.

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3. Application to Functions Defined by Fractional Derivatives

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

Definition 3.1. (See [5,6]; see also [11,12]). Let the functionf(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)^λ is removed by requiringlog(z−ζ)to be real whenz−ζ >0.

Using the above Definition3.1and its known extensions involving fractional derivatives and fractional integrals, Owa and Srivastava [5] introduced the op- eratorΩ^λ :A → Adefined by

(Ω^λf)(z) = Γ(2−λ)z^λD^λ_zf(z), (λ6= 2,3,4, . . .).

The classM_α,b^λ (φ)consists of functionsf ∈ Afor whichΩ^λf ∈M_α,b(φ). Note thatM_0,b⁰ (φ) = S_b^∗(φ)andM_0,1⁰ (φ) = S^∗(φ). AlsoM_α,b^λ (φ)is the special case of the classM_α,b^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,b^g (φ) if and only if

(f∗g)(z) = z+

∞

X

n=2

g_na_nzⁿ∈M_α,b(φ),

we obtain the coefficient estimate for functions in the class M_α,b^g (φ), from the corresponding estimate for functions in the class M_α,b(φ). Applying Theo- rem 2.1 for the function(f ∗g)(z) = z +g₂a₂z² +g₃a₃z³ +· · ·, we get the following theorem after 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,b^g (φ), then

|a₃ −µa²₂|

≤ B₁|b|

2g₃(1 + 3α)max

1,

B₂ B₁ +

(1 + 2α)g₂²−2µ(1 + 3α)g₃ (1 + 2α)²g₂²

bB₁

Since

(Ω^λf)(z) =z+

∞

X

n=2

Γ(n+ 1)Γ(2−λ) Γ(n+ 1−λ) anzⁿ,

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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,b^λ (φ), then

|a₃−µa²₂| ≤ B1|b|(2−λ)(3−λ) 12(1 + 3α)

×max

1,

B₂ B₁ +

(1 + 2α)(3−λ)−3µ(1 + 3α)(2−λ) (3−λ)(1 + 2α)² bB1

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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] F.R. KEOGH AND E.P. MERKES, A coefficient inquality for certain classes of analytic functions, Proc. Amer. Math. Soc., 20 (1969), 8–12.

[4] 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.

[5] S. OWA ANDH.M. SRIVASTAVA, Univalent and starlike generalized hy- pergeometric functions, Canad. J. Math., 39 (1987), 1057–1077.

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

[7] V. RAVICHANDRAN, METIN BOLCAL, YASAR POLATOGLU AND

A. SEN, Certian subclasses of Starlike and Convex functions of Complex Order, Hacettepe Journal of Mathematics and Statistics, 34 (2005), 9-15.

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

[9] T.N. SHANMUGAM, S. SIVASUBRAMANIAN AND M. DARUS, Fekete-Szegö inequality for a certain class of analytic functions, Preprint.

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[10] 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.

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

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