Also, certain applications of our results for a class of functions defined by convolution are given

http://jipam.vu.edu.au/

Volume 7, Issue 3, Article 117, 2006

FEKETE-SZEGÖ FUNCTIONAL FOR SOME SUBCLASS OF NON-BAZILEVI ˇC FUNCTIONS

T.N. SHANMUGAM, M.P. JEYARAMAN, AND S. SIVASUBRAMANIAN DEPARTMENT OFMATHEMATICS

ANNAUNIVERSITY

CHENNAI600025 TAMILNADU, INDIA

shan@annauniv.edu DEPARTMENT OFMATHEMATICS

EASWARIENGINEERINGCOLLEGE

CHENNAI-600 089 TAMILNADU, INDIA

jeyaraman mp@yahoo.co.in sivasaisastha@rediffmail.com

Received 18 November, 2005; accepted 24 March, 2006 Communicated by A. Lupa¸s

ABSTRACT. In this present investigation, the authors obtain a sharp Fekete-Szegö’s inequality for certain normalized analytic functions f(z) defined on the open unit disk for which (1 +β)

z f(z)

α

−βf⁰(z)

z f(z)

1+α

,(β ∈ C, 0 < α < 1) lies in a region starlike with respect to1 and is symmetric with respect to the real axis. Also, certain applications of our results 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 also obtained.

Key words and phrases: Analytic functions, Starlike functions, Subordination, Coefficient problem, Fekete-Szegö inequality.

2000 Mathematics Subject Classification. Primary 30C45.

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})

andS be the subclass ofAconsisting of univalent functions. Letφ(z)be an analytic function with positive real part on∆withφ(0) = 1,φ⁰(0)>0which maps the unit disk∆onto a region

ISSN (electronic): 1443-5756

c 2006 Victoria University. All rights reserved.

108-06

starlike with respect to1and is symmetric with respect to the real axis. LetS^∗(φ)be the class of functionsf ∈ S for which

zf⁰(z)

f(z) ≺φ(z), (z ∈∆) andC(φ)be the class of functionsf ∈ 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 [3]. They have obtained the Fekete-Szegö inequality for the functions in the classC(φ). Since f ∈ C(φ)if and only if zf⁰ ∈ S^∗(φ), we get the Fekete- Szegö inequality for functions in the classS^∗(φ). Recently, Shanmugam and Sivasubramanian [9] obtained Fekete- Szegö inequalities for the class of functionsf ∈ Asuch that

zf⁰(z) +αz²f⁰⁰(z)

(1−α)f(z) +αzf⁰(z) ≺φ(z) (0≤α <1).

Also, Ravichandran et al. [7] obtained the Fekete-Szegö inequality for the class of Bazileviˇc functions. 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. [11]. Obradovic [4]

introduced a class of functionsf ∈ A,such that, for0< α <1,

<

f⁰(z)

z f(z)

α

>0, z ∈∆.

He called this class of function as "Non-Bazileviˇc" type. Tuneski and Darus [14] obtained the Fekete-Szegö inequality for the non-Bazileviˇc class of functions. Using this non-Bazileviˇc class, Wang et al.[15] studied many subordination results for the classN(α, β, A, B)defined as

N(α, β, A, B) :=

(

f ∈ A : (1 +β) z

f(z) α

−βf⁰(z) z

f(z) 1+α

≺ 1 +Az 1 +Bz

) , whereβ ∈C,−1≤B ≤1, A6=B, 0< α <1.

In the present paper, we obtain the Fekete-Szegö inequality for functions in a more general class N_α,β(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 classN_α,β^λ (φ)of functions defined by fractional derivatives. The aim of this paper is to give a generalization the Fekete-Szegö inequalities for some subclass of Non-Bazileviˇc functions .

Definition 1.1. Letφ(z)be an 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 classN_α,β(φ)if

(1 +β) z

f(z) α

−βf⁰(z) z

f(z) 1+α

≺φ(z), (β ∈C,0< α <1).

For fixed g ∈ A, we define the classN_α,β^g (φ) to be the class of functions f ∈ A for which (f∗g)∈N_α,β(φ).

Remark 1.1. N_α,−1 ^1+z_1−z

is the class of Non-Bazileviˇc functions introduced by Obradovic [4].

Remark 1.2. N_{α,−11+(1−2γ)z}

1−z

, 0≤γ <1is the class of Non-Bazileviˇc functions of order γ introduced and studied by Tuneski and Darus [14].

Remark 1.3. We callN_α,β

1 + _π²2

log ¹⁺

√z 1−√

z

2

the class of "Non-Bazileviˇc parabolic star- like functions".

To prove our main result, we need the following:

Lemma 1.4 ([3]). Ifp₁(z) = 1 +c₁z+c₂z²+· · · is an analytic function with a positive real part in∆, then

|c₂ −vc²₁| ≤











−4v+ 2 ifv ≤0, 2 if0≤v ≤1, 4v−2 ifv ≥1.

Whenv < 0 orv > 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)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. Ifv = 1, the equality holds if and only ifp₁ is the reciprocal of one of the functions such that equality holds in the case ofv = 0.

Also the above upper bound is sharp, and it can be improved as follows when0< v < 1:

|c₂−vc²₁|+v|c₁|² ≤2 (0< v≤1/2) and

|c₂−vc²₁|+ (1−v)|c₁|² ≤2 (1/2< v ≤1).

2. FEKETE-SZEGÖPROBLEM

Our main result is the following:

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

|a₃−µa²₂| ≤















−_(α+2β)^B2 − _2(α+β)^µB12 2 +_2(α+β)^(1+α)2B₁² if µ≤σ₁;

−_(α+2β)^B1 if σ₁ ≤µ≤σ₂;

B2

(α+2β) + _2(α+β)^µB21 2 −_2(α+β)^(1+α)2B₁² if µ≥σ₂, where,

σ₁ := (1 +α)(2β+α)B₁²−2(B₂−B₁)(β+α)²

2(2β+α)B₁² ,

σ₂ := (1 +α)(2β+α)B₁²−2(B₂+B₁)(β+α)²

2(2β+α)B²₁ .

The result is sharp.

Proof. Forf ∈N_α,β(φ),let (2.1) p(z) := (1 +β)

z f(z)

α

−βf⁰(z) z

f(z) 1+α

= 1 +b₁z+b₂z²+· · · . From (2.1), we obtain

−(α+β)a₂ =b₁

(2β+α)

α+ 1

2 a²₂−a₃

=b₂. 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

a₃−µa²₂ =− B1

2(2β+α)

c₂−vc²₁ where

v := 1 2

1− B₂ B1

+(2β+α)(α+ 1−2µ) 2(β+α)² B₁

.

Our result now follows by an application of Lemma 1.4. To show that the bounds are sharp, we define the functionsK_α,β^φn (n = 2,3, . . .)by

(1 +β) z K_α,β^φn(z)

!α

−β K^φ_α,βⁿ 0

(z) z

K_α,β^φn(z)

!1+α

=φ(zⁿ⁻¹), K_α,β^φn(0) = 0 = [K_α,β^φn]⁰(0)−1

and the functionF_α,β^λ andG^λ_α,β (0< α <1)by (1 +β) z

F_α,β^λ (z)

!α

−β[F_α,β^λ ]⁰(z) z F_α,β^λ (z)

!1+α

=φ(zⁿ⁻¹), [F_α,β^λ ](0) = 0 = [F_α,β^λ ]]⁰(0)−1

and

(1 +β) z G^λ_α,β

!α

−β[G^λ_α,β]⁰(z) z G^λ_α,β(z)

!1+α

=φ(zⁿ⁻¹), [G^λ_α,β](0) = 0 = [G^λ_α,β]⁰(0)−1.

Clearly, the functionsK_α,β^φn,[F_α,β^λ ]and[G^λ_α,β]∈N_α,β(φ) Also we writeK_α,β^φ :=K_α,β^φ2 .

Ifµ < σ₁ orµ > σ₂, then the equality holds if and only iff isK_α,β^φ or one of its rotations.

Whenσ₁ < µ < σ₂, the equality holds if and only iff isK_α,β^φ3 or one of its rotations. Ifµ=σ₁ then the equality holds if and only iff isF_α,β^λ or one of its rotations. Ifµ=σ2 then the equality

holds if and only iff isG^λ_α,β or one of its rotations.

Corollary 2.2. Letφ(z) = 1 +_π²2

log¹⁺

√z 1−√

z

2

.Iff given by (1.1) belongs toN_α,β(φ), then

|a₃−µa²₂| ≤















−_3π2(α+2β)⁸ − _π4(α+β)^{8µ 2} + _(α+β)^(1+α)2

8

π⁴ if µ≤σ₁

−_π2(α+2β)⁴ if σ₁ ≤µ≤σ₂

8

3π²(α+2β) +_π4(α+β)^{8µ 2} − _(α+β)^(1+α)2

8

π⁴ if µ≥σ₂ where,

σ₁ := (1 +α)(2β+α)_π¹⁶4 −2 _3π⁸2 − _π⁴2

(β+α)² 2(2β+α)_π¹⁶4

σ₂ := (1 +α)(2β+α)_π¹⁶4 −2 _3π⁸2 + _π⁴2

(β+α)² 2(2β+α)_π¹⁶4

. The result is sharp.

Corollary 2.3. For β = −1, φ(z) = ^1+(1−2γ)z_1−z , 0 ≤ γ < 1 in Theorem 2.1, we get the results obtained by Tuneski and Darus [14].

Remark 2.4. Ifσ₁ ≤µ≤σ₂, then, in view of Lemma 1.4, Theorem 2.1 can be improved. Let σ₃ be given by

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

|a₃−µa²₂| − (β+α)² (2β+α)B₁²

B₁−B₂+B₁²(α+ 1−2µ)(2β+α) 2(β+α)²

|a₂|² ≤ − B₁ (2β+α). Ifσ₃ ≤µ≤σ₂, then

|a₃−µa²₂| − (β+α)² (2β+α)B₁²

B₁+B₂−B₁²(α+ 1−2µ)(2β+α) 2(β+α)²

|a₂|² ≤ − B₁ (2β+α). 3. APPLICATIONS TO FUNCTIONS DEFINED BYFRACTIONALDERIVATIVES

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

Definition 3.1 (see [5, 6]; see also [12, 13]). Letf 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 Definition 3.1 and its known extensions involving fractional derivatives and fractional integrals, Owa and Srivastava [5] introduced the operatorΩ^λ :A → Adefined by

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

The classN_α,β^λ (φ)consists of functionsf ∈ Afor whichΩ^λf ∈N_α,β(φ). Note thatN_α,β^λ (φ) is the special case of the classN_α,β^g (φ)when

(3.1) g(z) = z+

∞

X

n=2

Γ(n+ 1)Γ(2−λ) Γ(n+ 1−λ) zⁿ.

Letg(z) = z+P∞

n=2g_nzⁿ(g_n >0). Sincef(z) =z+P∞

n=2a_nzⁿ ∈N_α,β^g (φ)if and only if (f ∗g) =z +P∞

n=2g_na_nzⁿ ∈N_α,β(φ), we obtain the coefficient estimate for functions in the class N_α,β^g (φ), from the corresponding estimate for functions in the class N_α,β(φ). Applying Theorem 2.1 for the function (f ∗g)(z) = z +g₂a₂z² +g₃a₃z³ +· · ·, we get the following Theorem 3.1 after an obvious change of the parameterµ:

Theorem 3.1. Let the functionφbe given byφ(z) = 1 +B₁z+B₂z²+B₃z³+· · ·. Iffgiven by (1.1) belongs toN_α,β^g (φ), then

|a₃−µa²₂| ≤















1 g3

n−_(α+2β)^B2 − _g2^µg3B21

22(α+β)² +_2(α+β)^(1+α)2B₁²o

if µ≤σ₁;

−_g¹

3

B1

(α+2β) if σ₁ ≤µ≤σ₂;

1 g3

n B2

(α+2β) + _2(α+β)^µg3B122g²₂ − _2(α+β)^(1+α)2B₁²o

if µ≥σ₂, where

σ₁ := g₃ g₂²

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

σ₂ := g₃ g₂²

(1 +α)(2β+α)B₁²−2(B₂+B₁)(β+α)²

2(2β+α)B²₁ .

The result is sharp.

Since

(Ω^λf)(z) = z+

∞

X

n=2

Γ(n+ 1)Γ(2−λ) Γ(n+ 1−λ) anzⁿ, 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).

Forg₂andg₃given by (3.2) and (3.3), Theorem 3.1 reduces to the following:

Theorem 3.2. Let the functionφbe given byφ(z) = 1 +B1z+B2z²+B3z³+· · ·. Iffgiven by (1.1) belongs toN_α,β^g (φ), then

|a₃−µa²₂| ≤















(2−λ)(3−λ) 6

n

−_(α+2β)^B2 −_g2^µg3B12

22(α+β)² +_2(α+β)^(1+α)2B₁² o

if µ≤σ1;

−^{(2−λ)(3−λ)}_{6 (α+2β)}^B1 if σ₁ ≤µ≤σ₂;

(2−λ)(3−λ) 6

n B2

(α+2β) +_2(α+β)^µg3B21_2g2 2

−_2(α+β)^(1+α)2B₁²o

if µ≥σ₂, where

σ1 := 2(3−λ) 3(2−λ)

(1 +α)(2β+α)B₁²−2(B₂ −B₁)(β+α)²

2(2β+α)B₁² ,

σ₂ := 2(3−λ) 3(2−λ)

(1 +α)(2β+α)B₁²−2(B₂ +B₁)(β+α)²

2(2β+α)B₁² .

The result is sharp.

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