JJ II

(1)

volume 7, issue 3, article 117, 2006.

Received 18 November, 2005;

accepted 24 March, 2006.

Communicated by:A. Lupa¸s

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

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

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

Department of Mathematics Anna University

Chennai 600025 Tamilnadu, India

EMail:shan@annauniv.edu Department of Mathematics Easwari Engineering College Chennai-600 089

Tamilnadu, India

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

c

2000Victoria University ISSN (electronic): 1443-5756 108-06

(2)

Fekete-Szegö Functional for some Subclass of Non-Bazilevi ˇc

Functions

T.N. Shanmugam, M.P. Jeyaraman and S. Sivasubramanian

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of16

J. Ineq. Pure and Appl. Math. 7(3) Art. 117, 2006

Abstract

In this present investigation, the authors obtain a sharp Fekete-Szegö’s inequality for certain normalized analytic functionsf(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 to1and 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.

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 to1and is symmetric with respect to the real axis. Let S^∗(φ) be the class of functions f ∈ 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 [3]. They have obtained the Fekete-Szegö inequality for the functions in the classC(φ). Sincef ∈ C(φ)if and only ifzf⁰ ∈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).

(4)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of16

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 functions f ∈ 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 classNα,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 class N_α,β^λ (φ)of functions defined by fractional derivatives. The aim of this paper is to give a generaliza- tion the Fekete-Szegö inequalities for some subclass of Non-Bazileviˇc functions .

(5)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of16

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

Remark 1. Nα,−1 1+z 1−z

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

Remark 2. Nα,−1

1+(1−2γ)z 1−z

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

Remark 3. We call N_α,β

1 + _π²2

log ¹⁺

√z 1−√

z

2

the class of "Non-Bazileviˇc parabolic starlike functions".

To prove our main result, we need the following:

Lemma 1.1 ([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.

(6)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of16

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) 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 equality holds in the case ofv = 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).

(7)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of16

2. Fekete-Szegö Problem

Our main result is the following:

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

|a₃−µa²₂| ≤











−_(α+2β)^B² − _2(α+β)^µB²¹ 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)B₁² . The result is sharp.

Proof. Forf ∈Nα,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

(8)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of16

(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

.

(9)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of16

Our result now follows by an application of Lemma 1.1. 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,βⁿ⁰

(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]∈Nα,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. When σ₁ < 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

(10)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of16

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

log ¹⁺

√z 1−√

z

2

.Iff given by (1.1) belongs to N_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)¹⁶_π4 −2 _3π⁸2 + _π⁴2

(β+α)² 2(2β+α)¹⁶_π4

. The result is sharp.

Corollary 2.3. Forβ =−1, φ(z) = ^1+(1−2γ)z_1−z , 0≤γ <1in Theorem2.1, we get the results obtained by Tuneski and Darus [14].

Remark 4. If σ₁ ≤ µ ≤ σ₂, then, in view of Lemma 1.1, Theorem2.1 can be improved. Letσ₃be given by

σ₃ := (1 +α)(2β+α)B₁²−2B₂(β+α)² 2(2β+α)B₁² .

(11)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of16

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).

(12)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of16

3. Applications to Functions Defined by Fractional Derivatives

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

Definition 3.1 (see [5, 6]; see also [12, 13]). Let f be analytic in a simply connected region of thez-plane containing the origin. The fractional derivative off of orderλis defined by

D^λ_zf(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 [5] 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 N_α,β^λ (φ)consists of functionsf ∈ A for whichΩ^λf ∈ Nα,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. Let g(z) = z +P∞

n=2g_nzⁿ (g_n > 0). Since f(z) = z +P∞

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

n=2gnanzⁿ ∈ Nα,β(φ), we obtain the

(13)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of16

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

Theorem 3.1. Let the functionφbe given byφ(z) = 1 +B1z+B2z²+B3z³+

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

|a₃−µa²₂| ≤











1 g3

n−_(α+2β)^B² − _g2^µg³^B¹²

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)^µg³^B¹2²g²₂ − _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)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,

(14)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of16

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 functionφbe given byφ(z) = 1 +B₁z+B₂z²+B₃z³+

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

|a₃−µa²₂|

≤











(2−λ)(3−λ) 6

n

−_(α+2β)^B² −_g2^µg³^B¹²

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)^µg³^B¹2²g²₂ − _2(α+β)^(1+α)2B₁² o

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.

(15)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of16

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

169.

[4] M. OBRADOVIC, A class of univalent functions, Hokkaido Math. J., 27(2) (1998), 329–335.

[5] S. OWA AND H.M. SRIVASTAVA, Univalent and starlike generalized bypergeometric 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, A. GANGADHARAN ANDM. DARUS, Fekete- Szeg˝o inequality for certain class of Bazilevic functions, Far East J. Math.

Sci. (FJMS), 15(2) (2004), 171–180.

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

(16)

Functions

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of16

[9] T.N. SHANMUGAM AND S. SIVASUBRAMANIAN, On the Fekete- Szegö problem for some subclasses of analytic functions, J. Inequal. Pure and Appl. Math., 6(3) (2005), Art. 71, 6 pp.

[10] 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.

[11] 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.

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

[13] 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).

[14] N. TUNESKI AND M. DARUS, Fekete-Szegö functional for non- Bazilevic functions, Acta Mathematica Academia Paedagogicae Nyiregy- haziensis, 18 (2002), 63–65.

[15] Z. WANG, C. GAO AND M. LIAO, On certain generalized class of non- Bazileviˇc functions, Acta Mathematica Academia Paedagogicae Nyiregy- haziensis, 21 (2005), 147–154.