Kyungpook Mathematical Journal 2018; 58(4): 677-688  
Coefficient Estimates for a Subclass of Bi-univalent Functions Defined by Sălăgean Type q-Calculus Operator
Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, India, e-mail : kamble.prakash69@gmail.com Mallikarjun, Department of Mathematics, Dr. D Y Patil School of Engineering and Technology, Pune 412205, India, e-mail : mgshrigan@gmail.com
*Corresponding Author.
Received: March 7, 2018; Revised: September 25, 2018; Accepted: October 2, 2018; Published online: December 23, 2018.
© Kyungpook Mathematical Journal. All rights reserved.

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Abstract

In this paper, we introduce and investigate a new subclass of bi-univalent functions defined by Sălăgean q-calculus operator in the open disk . For functions belonging to the subclass, we obtain estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3|. Some consequences of the main results are also observed.

Keywords: analytic functions, bi-univalent functions, coefficient bounds, Sălăgean q-differential operator , Sălăgean derivative.
1. Introduction

Let denote the family of functions analytic in the open unit disk

U={z:z         and         z<1},

which are normalized by the condition:

f(0)=f(0)-1=0

and given by the following Taylor-Maclaurin series:

f(z)=z+k=2akzk.

Also let be the class of functions of the form given by (1.1), which are univalent in . The Koebe one-quarter theorem [7] ensures that the image of under every univalent function contains a disk of radius 14. Hence every function has an inverse f−1, defined by

f-1(f(z))=z,         (zU)

and

f-1(f(w))=w,         (w<r0(f);r0(f)14),

where

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

A function is said to be bi-univalent in if both f and f−1 are univalent in . Let ∑ denote the class of bi-univalent functions in given by the Taylor-Maclaurin series expansion (1.1). For a brief history and interesting examples of functions in the class ∑, see [28] (see also [4]). From the work of Srivastava et al. [28], we choose to recall the following examples of functions in the class ∑:

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

However, familiar Koebe function is not a member of ∑.

The class of bi-univalent functions was investigated by Lewin [13], who proved that |a2| < 1.51. In 1981, Styer and Wright [30] showed that |a2| > 4/3. Subsequently, Brannan and Clunie [3] improved Lewin’s result to a22. Netanyahu [14], showed that maxfΣa2=43. In 1985, Branges [2] proved Bieberbach conjecture which showed that

ann;(nN-1),

N being positive integer.

The problem of finding coefficient estimates for the bi-univalent functions has received much attention in recent years. In fact, the aforecited work of Srivastava et al. [28] essentially revived the investigation of various subclasses of bi-univalent function class ∑ in recent years and that it leads to a flood of papers on the subject (see, for e.g., [6, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29]); it was followed by such works as those by Tang et al . [31], Xu et al . [32, 33] and Lashin [12], and others (see, for e.g., [1, 5, 8]). The coefficient estimate problem involving the bound of |an|(n ∈ ℕ {1, 2}) for each f ∈ ∑ is still an open problem.

In the field of geometric function theory, various subclasses of the normalized analytic function class have been studied from different view points. The q-calculus as well as the fractional calculus provide important tools that have been used in order to investigate various subclasses of . Historically speaking, the firm footing of the usage of the q-calculus in the context of geometric function theory which was actually provided and q-hypergeometric functions were first used in geometric function theory in a book chapter by Srivastava (see, for details, [18, pp. 347 et seq.]). Ismail et al. [10] introduced the class of generalized complex functions via q-calculus on some subclasses of analytic functions. Recently, Purohit and Raina [16] investigated applications of fractional q-calculus operator to define new classes of functions which are analytic in unit disk (see, for details, [9]).

For 0 < q < 1, the q-derivative of a function f given by (1.1) is defined as

Dqf(z)={f(qz)-f(z)(q-1)zfor z0,f(0)for z=0.

We note that limq1-Dqf(z)=f(z). From (1.2), we deduce that

Dqf(z)=1+k=2[k]qakzk-1,

where as q → 1

[k]q=1-qk1-q=1+q++qkk.

Making use of the q-differential operator for function , we introduced the Sălăgean q-differential operator as given below

Dq0f(z)=f(z)Dq1f(z)=zDqf(z)Dqnf(z)=zDq(Dqn-1f(z))Dqnf(z)=z+k=2[k]qnakzk         (n0,zU).

We note that limq → 1

Dnf(z)=z+k=2knakzk         (n0,zU),

the familiar Sălăgean derivative [17].

Recently, Kamble and Shrigan [11] introduce the following two subclasses of the bi-univalent function class ∑ and obtained estimate on first two Taylor-Maclaurin coefficients |a2| and |a3| for functions in these subclasses as follows.

Definition 1.1.([11])

For 0 < α ≤ 1, 0 < q < 1, λ ≥ 1, μ ≥ 0 and n ∈ ℕ0, a function f(z) given by (1.1) is said to be in the classΣq,μ(n,α,λ) if the following conditions are satisfied

fΣand |arg((1-λ)(Dqnf(z)z)μ+λ(Dqnf(z))(Dqnf(z)z)μ-1)|<απ2

and

|arg((1-λ)(Dqng(w)w)μ+λ(Dqng(w))(Dqng(w)w)μ-1)|<απ2,

where the function g is given by

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

and Dqn is the Sălăgean q-differential operator.

Theorem 1.2.([11])

Let f(z) given by (1.1) be in the function classΣq,μ(n,α,λ). Then

a22αα(2(2λ+μ)[3]qn-(λ2+2λ+μ)[2]q2n)+(λ+μ)2[2]q2n

and

a34α2(λ+μ)2[2]q2n+2α(2λ+μ)[3]qn,

where 0 < α ≤ 1, 0 < q < 1, λ ≥ 1, μ ≥ 0 and n ∈ ℕ0.

Definition 1.3.([11])

For 0 ≤ β < 1, 0 < q < 1, λ ≥ 1, μ ≥ 0 and n ∈ ℕ0, a function f(z) given by (1.1) is said to be in the classΣq,μ(n,β,λ) if the following conditions are satisfied

fΣand Re{(1-λ)(Dqnf(z)z)μ+λ(Dqnf(z))(Dqnf(z)z)μ-1}>β

and

Re{(1-λ)(Dqng(w)w)μ+λ(Dqng(w))(Dqng(w)w)μ-1}>β.

Theorem 1.4.([11])

Let f(z) given by (1.1) be in the function classΣq,μ(n,β,λ). Then

a2min{4(1-β)2[3]qn+(μ-1)[2]q2n(2λ+μ),2(1-β)(λ+μ)[2]qn}

and

a3min{4(1-β)2(λ+μ)2[2]q2n+2(1-β)(2λ+μ)[3]qn,(1-β){4[3]qn+[2]q2n(μ-1)-[2]q2n(μ-1)}2[3]qn+(μ-1)[2]q2n(2λ+μ)[3]qn},

where 0 ≤ β < 1, 0 < q < 1, λ ≥ 1, μ ≥ 0 and n ∈ ℕ0.

Remark 1.5

By appropriately specializing the parameters in Definition 1.1 and 1.3, we can get several known subclasses of the bi-univalent function class ∑. For example:

For n = 0 and q → 1, we obtain the bi-univalent function classes Σ1,μ(0,α,λ)=NΣμ(α,λ)         and         Σ1,μ(0,β,λ)=NΣμ(β,λ)(see [ 21]);

For μ = 1, n = 0 and q → 1, we obtain the bi-univalent function classes Σ1,1(0,α,λ)=Σ(α,λ)         and         Σ1,1(0,β,λ)=Σ(β,λ)(see [ 8]);

For μ = 1 and q → 1 we obtain the bi-univalent function classes Σ1,1(n,α,λ)=Σ(n,α,λ)         and         Σ1,1(n,β,λ)=Σ(n,β,λ)(see [ 15]);

For μ = 1, n = 0, λ = 1 and q → 1, we obtain the bi-univalent function classes Σ1,1(0,α,1)=Σα         and         Σ1,1(0,β,1)=Σ(β)(see [ 28]);

For μ = 0, n = 0, λ = 1 and q → 1, we obtain the bi-univalent function classes Σ1,0(0,α,1)=SΣ*(α)         and         Σ1,0(0,β,1)=SΣ*(β)(see [ 4]);

This paper is a sequel to some of the aforecited works (especially see [11, 32, 33]). Here we introduce and investigate the general subclass Σh,p(λ,μ,n,q)(0<q<1,λ1,μ0) of the analytic function class , which is given by Definition 1.6 below.

Definition 1.6

Let h, be analytic functions and

min{Re(h(z)),Re(p(z))}>0         (zU)         and         h(0)=p(0)=1.

Also let the function f given by (1.1), be in the analytic function class . We say thatfΣh,p(λ,μ,n,q)         (0<q<1,λ1,μ0and n0)

if the following conditions satisfied:

fΣand (1-λ)(Dqnf(z)z)μ+λ(Dqnf(z))(Dqnf(z)z)μ-1h(U)   (zU)

and

(1-λ)(Dqng(w)w)μ+λ(Dqng(w))(Dqng(w)w)μ-1p(U)   (wU),

where the function g is given by (1.9).

If fΣh,p(λ,μ,n,q), then

fΣand |arg((1-λ)(Dqnf(z)z)μ+λ(Dqnf(z))(Dqnf(z)z)μ-1)|<απ2

and

|arg((1-λ)(Dqng(w)w)μ+λ(Dqng(w))(Dqng(w)w)μ-1)|<απ2

or

fΣand Re{(1-λ)(Dqnf(z)z)μ+λ(Dqnf(z))(Dqnf(z)z)μ-1}>β

and

Re{(1-λ)(Dqng(w)w)μ+λ(Dqng(w))(Dqng(w)w)μ-1}>β.

where the function g is given by (1.9).

Our paper is motivated and stimulated especially by the work of Srivastava et al. [21, 28]. Here we propose to investigate the bi-univalent function subclass Σh,p(λ,μ,n,q) of the function class ∑ and find estimates on the initial coefficients |a2| and |a3| for functions in the new subclass of the function class ∑ using Sălăgean q-differential operator.

2. A Set of General Coefficient Estimates

In this section, we derive estimates on the initial coefficients |a2| and |a3| for functions in subclass Σh,p(λ,μ,n,q) given by Definition 1.6.

Theorem 2.1

Let the function f(z) given by Taylor-Maclaurin series expansion (1.1) be in the function classΣh,p(λ,μ,n,q). Then

a2min{h(0)2+p(0)22(λ+μ)2[2]q2n,h(0)+p(0)2(2λ+μ)(μ-1)[2]q2n+2[3]qn}

and

a3min{h(0)2+p(0)22(λ+μ)2[2]q2n+h(0)+p(0)4(2λ+μ)[3]qn,(μ-1)[2]q2n+4[3]qnh(0)+μ-1[2]q2np(0)4(2λ+μ)[3]qn(μ-1)[2]q2n+2[3]qn},

where 0 < q < 1, λ ≥ 1, μ ≥ 0 and n ∈ ℕ0.

Proof

It follows from (1.16) and (1.17) that

(1-λ)(Dqnf(z)z)μ+λ(Dqnf(z))(Dqnf(z)z)μ-1=h(U)

and

(1-λ)(Dqng(w)w)μ+λ(Dqng(w))(Dqng(w)w)μ-1=p(U)

Comparing the coefficients of z and z2 in (2.3) and (2.4), we have

(λ+μ)[2]qna2=h1,(μ-1)(λ+μ2)[2]q2na22+(2λ+μ)[3]qna3=h2,-(λ+μ)[2]qna2=p1

and

-(2λ+μ)[3]qna3+(4[3]qn+(μ-1)[2]q2n)   (λ+μ2)   a22=p2.

From (2.5) and (2.7), we obtain

h1=-p1

and

2(λ+μ)2[2]q2na22=h12+p12.

Also, from (2.6) and (2.8), we find that

{(μ-1)[2]q2n+2[3]qn}   (2λ+μ)2a22=h2+p2.

Therefore, we find from the equations (2.10) and (2.11) that

a2h(0)2+p(0)22(λ+μ)2[2]q2n

and

a2h(0)+p(0)2(2λ+μ)(μ-1)[2]q2n+2[3]qn,

respectively. So we get the desired estimate on the coefficients |a2| as asserted in (2.1).

Next, in order to find the bound on the coefficient |a3|, we subtract (2.8) from (2.6), we get

2(2λ+μ)[3]qna3-2[3]qn(2λ+μ)a22=h2-p2.

Upon substituting the value of a22 from (2.10) into (2.12), we arrive at

a3=h12+p122(λ+μ)2[2]q2n+h2-p22(2λ+μ)[3]qn.

We thus find that

a3h(0)2+p(0)22(λ+μ)2[2]q2n+h(0)+p(0)4(2λ+μ)[3]qn.

On the other hand, upon substituting the value of a22 from (2.11) into (2.12), we arrive at

a3={(μ-1)[2]q2n+4[3]qn}h2+(μ-1)[2]q2np22(2λ+μ)[3]qn{(μ-1)[2]q2n+2[3]qn}.

Consequently, we have

a3(μ-1)[2]q2n+4[3]qnh(0)+μ-1[2]q2np(0)4(2λ+μ)[3]qn(μ-1)[2]q2n+2[3]qn.

This evidently completes the proof of Theorem 2.1.

3. Corollaries and Consequences

By Setting μ = 1, q → 1 and n = 0 in Theorem 2.1, we deduce the following consequence of Theorem 2.1.

Corollary 3.1

Let the function f(z) given by Taylor-Maclaurin series expansion (1.1) be in the function classΣh,p(λ)(λ1). Then

a2min{h(0)2+p(0)22(1+λ)2,h(0)+p(0)4(1+2λ)}

and

a3min{h(0)2+p(0)22(1+λ)2+h(0)+p(0)4(1+2λ),h(0)2(1+2λ)}.

By Setting μ = 0, λ = 1, q → 1 and n = 0 in Theorem 2.1, we deduce the following.

Corollary 3.2.([5])

Let the function f(z) given by Taylor-Maclaurin series expansion (1.1) be in the function classΣh,p. Then

a2min{h(0)2+p(0)22,h(0)+p(0)4}

and

a3min{h(0)2+p(0)28+h(0)+p(0)8,3h(0)+p(0)8}.

Remark 3.3

Corollary 3.2 is an improvement of the following estimates obtained by Xu et al . [33].

Corollary 3.4.([33])

Let the function f(z) given by Taylor-Maclaurin series expansion (1.1) be in the function classΣh,p(λ)(λ1). Then

a2h(0)+p(0)4(1+2λ)

and

a3h(0)2(1+2λ).

By Setting λ = 1, μ = 1, q → 1 and n = 0 in Theorem 2.1, we deduce the following Corollary 3.5.

Corollary 3.5.([32])

Let the function f(z) given by Taylor-Maclaurin series expansion (1.1) be in the function classΣh,p. Then

a2h(0)+p(0)12

and

a3h(0)6.
4. Concluding Remarks and Observations

The main objective in this paper has been to derive first two Taylor-Maclaurin coefficient estimates for functions belonging to a new subclass Σh,p(λ,μ,n,q) of analytic and bi-univalent function in the open unit disk . Indeed, by using Sălăgean q-calculus operator, we have successfully determined the first two Taylor-Maclaurin coefficient estimates for functions belonging to a new subclass Σh,p(λ,μ,n,q).

By means of corollaries and consequences which we discuss in the preceding section by suitable specializing the parameters λ and μ, we have also shown already that the results presented in this paper would generalize and improve some recent works of Xu et al . [32, 33] and other authors.

Acknowledgements

We thank the referees for their insightful suggestions and scholarly guidance to revise and improve the results as in present form.

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