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Kyungpook Mathematical Journal 2017; 57(4): 613-621

Published online December 23, 2017

Copyright © Kyungpook Mathematical Journal.

Coefficient Estimates for Sãlãgean Type λ-bi-pseudo-starlike Functions

Santosh Joshi1
Sahsene Altinkaya and Sibel Yalçin2

Department of Mathematics, Walchand College of Engineering, Sangli 416415, India1
Department of Mathematics, Faculty of Arts and Science, Uludag University, 16059, Bursa, Turkey2

Received: September 28, 2016; Revised: October 2, 2017; Accepted: October 30, 2017

In this paper, we have constructed subclasses of bi-univalent functions associated with λ–bi-pseudo-starlike functions in the unit disc U. Furthermore we established bound on the coefficients for the subclasses SΣλ(k,α) and SΣλ(k,β).

Keywords: analytic functions, bi-starlike functions, coefficient bounds

Let A denote the class of functions f which are analytic in the open unit disc U = {z : z ∈ ℂ and |z| < 1}, of the form

f(z)=z+n=2anzn.

Let S be the subclass of A consisting of the form (1.1) which are univalent in U. It is well known that every function fS has an inverse f−1, satisfying f−1 (f (z)) = z, (zU) and f (f−1 (w)) = w, (|w| < r0 (f),r0(f)14), where

f-1(w)=w-a2w2+(2a22-a3)w3-(5a23-5a2a3+a4)w4+.

A function fA is said to be bi-univalent in U if both f and f−1 are univalent in U. Let ∑ denote the class of bi-univalent functions defined in the unit disc U. For a brief history and interesting examples of functions in the class ∑, see the pioneering work on this area by Srivastava et al. [15], which has apparently revived the study of bi-univalent functions in recent years. From the work of Srivastava et al. [15], we recall the following examples of functions in the class ∑ :

z1-z,         -log(1-z),         12log(1+z1-z).

However, the familiar Koebe function is not a member of the bi-univalent function class ∑. Such other common examples of functions in S as

z-z22         and         z1-z2

are also not members of ∑ (see [15]).

Historically, Lewin [11] studied the class of bi-univalent functions, obtaining the bound 1.51 for the modulus of the second coefficient |a2|. Subsequently, Brannan and Clunie [4] conjectured that a22 for f ∈ ∑. Later on, Netanyahu [12] showed that maxa2=43 if f (z) ∈ ∑. Brannan and Taha [5] introduced certain subclasses of the bi-univalent function class ∑ similar to the familiar subclasses and of starlike and convex functions of order β (0 ≦ β < 1) in , respectively (see [12]). The classes SΣ*(β) and of bi-starlike functions of order β in and bi-convex functions of order β in , corresponding to the function classes and , were also introduced analogously. For each of the function classes SΣ*(β) and , they found non-sharp estimates for the initial coefficients. Recently, motivated substantially by the aforementioned work on this area Srivastava et al. [15], many authors investigated the coefficient bounds for various subclasses of bi-univalent functions (see, for example, [2, 6, 7, 17, 18, 19, 20, 21]). Not much is known about the bounds on the general coefficient |an| for n ≧ 4. In the literature, there are only a few works determining the general coefficient bounds for |an| for the analytic bi-univalent functions (see, for example, [1, 8, 9]). The coefficient estimate problem for each of the coefficients |an| (n ∈ ℕ {1, 2} ; ℕ = {1, 2, 3, · · · }) is still an open problem.

For f belongs to A, Sãlãgean (see [14]) defined differential operator Dk, k ∈ ℕ0 = ℕ ∪ {0}, by

D0f(z)=f(z);D1f(z)=Df(z)=zf(z);Dkf(z)=D(Dk-1f(z)).

We note that

Dkf(z)=z+n=2nkanzn.

In this paper, motivated by the earlier work of Babalola [3] and Joshi et. al. [10], we aim at introducing two new subclasses of the function class ∑ and find estimate on the coefficients |a2| and |a3| for functions in these new subclasses of the function class ∑ employing the techniques used earlier by Srivastava et al. [15] (see also [7, 16, 22, 23, 24]).

We note the following lemma required for obtaining our results.

Lemma 1.1.([13])

If p (z) = 1+p1z +p2z2 +p3z3 +· · · is an analytic function in U with positive real part, thenpn2         (n={1,2,})

and

|p2-p122|2-p122.

Definition 2.1

A function f ∈ ∑ is said to be in the classSΣλ(k,α) if the following conditions are satisfied:

|arg (z[(Dkf(z))]λDkf(z))|<απ2         (0<α1,λ1,zU)

and

|arg (w[(Dkg(w))]λDkg(w))|<απ2         (0<α1,λ1,wU)

where the function g = f−1.

Theorem 2.2

Let f given by (1.1) be in the classSΣλ(k,α), 0 < α ≤ 1. Then

a22α(2λ-1)(2λ+α-1)

and

a32α(3λ-1)+4α2(2λ-1)2.
Proof

Let fSΣλ(k,α). Then

z[(Dkf(z))]λDkf(z)=[p(z)]αw[(Dkg(w))]λDkg(w)=[q(w)]α

where g = f−1, p, q in P and have the forms

p(z)=1+p1z+p2z2+

and

q(w)=1+q1w+q2w2+.

Now, equating the coefficients in (2.3) and (2.4), we get

(2λ-1)2ka2=αp1,(3λ-1)3ka3+(2λ2-4λ+1)22ka22=αp2+α(α-1)2p12,

and

-(2λ-1)2ka2=αq1,(2λ2+2λ-1)22ka22-(3λ-1)3ka3=αq2+α(α-1)2q12.

From (2.5) and (2.7) we obtain

p1=-q1

and

(2λ-1)222k+1a22=α2(p12+q12).

Also from (2.6), (2.8) and (2.10) we have

(2λ2-λ)22k+1a22=α(p2+q2)+α(α-1)2(p12+q12)=α(p2+q2)+α(α-1)2(2λ-1)222k+1α2a22.

Therefore, we have

a22=α2(p2+q2)(2λ-1)(2λ+α-1)22k.

Applying Lemma 1.1 for the coefficients p2 and q2, we obtain

a221-kα(2λ-1)(2λ+α-1).

Next, in order to find the bound on |a3|, by subtracting (2.8) from (2.6), we obtain

2(3λ-1)3ka3-2(3λ-1)22ka22=α(p2-q2)+α(α-1)2(p12-q12).

Then, in view of (1.3) and (2.10), we have

a32α(3λ-1)3k+4α2(2λ-1)222k.

This completes the proof of Theorem 2.2.

Putting k = 0 in Theorem 2.2, we have

Remark 2.3.([10])

Let f given by (1.1) be in the class SΣλ(α), 0 < α ≤ 1. Then

a22α(2λ-1)(2λ+α-1)

and

a32α(3λ-1)+4α2(2λ-1)2.

Definition 3.1

A function f ∈ ∑ is said to be in the class SΣλ(k,β) if the following conditions are satisfied:

(z[(Dkf(z))]λDkf(z))>β         (0β<1,λ1,zU)

and

(w[(Dkg(w))]λDkg(w))>β         (0β<1,λ1,wU)

where the function g = f−1.

Theorem 3.2

Let f given by (1.1) be in the classSΣλ(k,β), 0 ≤ β < 1. Then

a22(1-β)λ(2λ-1)22k

and

a32(1-β)(3λ-1)3k+4(1-β)2(2λ-1)222k.
Proof

Let fSΣλ(k,β). Then

z[(Dkf(z))]λDkf(z)=β+(1-β)p(z)w[(Dkg(w))]λDkg(w)=β+(1-β)q(w)

where p, qP and g = f−1.

It follows from (3.3) and (3.4) that

(2λ-1)2ka2=(1-β)p1,(3λ-1)3ka3+(2λ2-4λ+1)22ka22=(1-β)p2,

and

-(2λ-1)2ka2=(1-β)q1,(2λ2+2λ-1)22ka22-(3λ-1)3ka3=(1-β)q2.

From (3.6) and (3.8) we obtain

p1=-q1

and

(2λ-1)222k+1a22=(1-β)2(p12+q12).

Also from (3.6), (3.8) and (3.9) we have

(2λ2-λ)22k+1a22=(1-β)(p2+q2).

Therefore, we have

a22=(1-β)(p2+q2)(2λ2-λ)22k+1.

Appyling Lemma 1.1 for the coefficients p2 and q2, we obtain

a22(1-β)λ(2λ-1)22k.

Next, in order to find the bound on |a3|, by subtracting (3.8) from (3.6), we obtain

2(3λ-1)3ka3-2(3λ-1)22ka22=(1-β)(p2-q2).

Then, in view of (1.3) and (3.11), we have

a32(1-β)(3λ-1)3k+4(1-β)2(2λ-1)222k.

This completes the proof of Theorem 3.2.

Putting k = 0 in Theorem 3.2, we have

Remark 3.3.([10])

Let f given by (1.1) be in the class SΣλ(β), 0 ≤ β < 1. Then

a22(1-β)λ(2λ-1)

and

a32(1-β)(3λ-1)+4(1-β)2(2λ-1)2.

Taking k = 0 and λ = 1 in Theorems 2.2 and 3.2 one can get the following corollaries.

Corollary 3.4

Let f given by (1.1) be in the class S(α), 0 < α ≤ 1. Then

a22αα+1

and

a3α+4α2.

Corollary 3.5

Let f given by (1.1) be in the class S(β), 0 ≤ β < 1. Then

a22(1-β)

and

a3(1-β)+4(1-β)2.
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