Coefficient Bounds for Certain Subclasses of Bi-Univalent Function

Abstract

We introduce two new subclasses of the function class Σ of bi-univalent functions defined in the open unit disc. Furthermore, we find estimates on the coefficients |𝑎₂| and |𝑎₃| for functions in these new subclasses. Also consequences of the results are pointed out.

1. Introduction

Denote by 𝒜 the class of functions of the form 𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛 (1) which are analytic in the open unit disc 𝕌={𝑧∶𝑧∈ℂand|𝑧|<1}. Further, denote by 𝒮 the class of all functions in 𝒜 which are univalent and normalized by 𝑓(0)=0=𝑓^′(0) in 𝕌. The well-investigated subclasses of the univalent function class 𝒮 are the class of starlike functions of order 𝛼(0≤𝛼<1), denoted by 𝒮^∗(𝛼) and the class of convex functions of order 𝛼 denoted by 𝒦(𝛼) in 𝕌. It is well known that every function 𝑓∈𝒮 has an inverse 𝑓⁻¹, defined by 𝑓⁻¹𝑓(𝑧𝑓𝑓)=𝑧,𝑧∈𝕌,₋₁(𝑤)=𝑤,|𝑤|<𝑟₀𝑓,𝑟₀𝑓≥14, (2) where 𝑓⁻¹(𝑤)=𝑤−𝑎₂𝑤²+2𝑎²₂−𝑎₃𝑤³−5𝑎³₂−5𝑎₂𝑎₃+𝑎₄𝑤⁴+⋯. (3)

A function 𝑓∈𝒜 is said to be bi-univalent in 𝕌 if both 𝑓(𝑧) and 𝑓⁻¹(𝑧) are univalent in 𝕌. Let Σ denote the class of bi-univalent functions in 𝕌 given by (1).

Analogous to the function class 𝒮, the bi-univalent function class Σ include, for example, the class 𝒮^∗_Σ(𝛼) of bi-starlike functions of order 𝛼  (0≤𝛼<1), the class 𝒦_Σ(𝛼) of biconvex functions of order 𝛼  (0≤𝛼<1), and the class 𝒮^𝛼_Σ of strongly bi-starlike functions of order 𝛼  (0<𝛼≤1). For some intriguing examples of functions and characterization of the class Σ, one could refer to Srivastava et al. [1] and the references stated therein (see also [2]). Recently there has been triggering, interest to study the bi-univalent functions class Σ (see [2–5]) and obtain nonsharp estimates on the first two Taylor-Maclaurin coefficients |𝑎₂| and |𝑎₃|. The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients |𝑎_𝑛| for 𝑛∈ℕ⧵{1,2}, ℕ∶={1,2,3,…} is presumably still an open problem.

Motivated by the earlier works of Srivastava et al. [1] and Frasin and Aouf [3] in the present paper we introduce the following two new subclasses of the function class Σ.

Definition 1. A function 𝑓(𝑧) given by (1) is said to be in the class 𝒢_Σ(𝛼,𝜆) if the following conditions are satisfied: 𝑓∈Σ,arg𝑧𝑓^′(𝑧)(1−𝜆)𝑓(𝑧)+𝜆𝑧𝑓^′(𝑧)<𝛼𝜋2,0<𝛼≤1,0≤𝜆<1,𝑧∈𝕌,arg𝑤𝑔^′(𝑤)(1−𝜆)𝑔(𝑤)+𝜆𝑤𝑔^′(𝑤)<𝛼𝜋2,0<𝛼≤1,0≤𝜆<1,𝑤∈𝕌, (4) where the function 𝑔 is given by 𝑔(𝑤)=𝑤−𝑎₂𝑤²+2𝑎²₂−𝑎₃𝑤³−5𝑎³₂−5𝑎₂𝑎₃+𝑎₄𝑤⁴+⋯. (5)

Definition 2. A function 𝑓(𝑧) given by (1) is said to be in the class ℳ_Σ(𝛽,𝜆) if the following conditions are satisfied: 𝑓∈Σ,ℜ𝑧𝑓^′(𝑧)(1−𝜆)𝑓(𝑧)+𝜆𝑧𝑓^′(𝑧)ℜ>𝛽,0≤𝛽<1,0≤𝜆<1,𝑧∈𝕌,𝑤𝑔^′(𝑤)(1−𝜆)𝑔(𝑤)+𝜆𝑤𝑔^′(𝑤)>𝛽,0≤𝛽<1,0≤𝜆<1,𝑤∈𝕌, (6) where the function 𝑔 is given by (5).
It is of interest to note that, for 𝜆=0, the class 𝒢_Σ(𝛼,𝜆) reduces to 𝒮^𝛼_Σ of strongly bi-starlike functions of order 𝛼 and the class ℳ_Σ(𝛽,𝜆) leads to 𝒮^∗_Σ(𝛽) bi-starlike functions of order 𝛽.
The object of the present paper is to find estimates on the coefficients |𝑎₂| and |𝑎₃| for functions in the above-defined subclasses 𝒢_Σ(𝛼,𝜆) and ℳ_Σ(𝛽,𝜆) of the function class Σ by employing the techniques used earlier by Srivastava et al. [1].
In order to derive our main results, we recall the following lemma.

Lemma 3 (see [6]). If ℎ∈𝒫, then |𝑐_𝑘|≤2 for each 𝑘, where 𝒫, is the family of all functions ℎ analytic in 𝕌 for which ℜ{ℎ(𝑧)}>0, where ℎ(𝑧)=1+𝑐₁𝑧+𝑐₂𝑧²+⋯ for 𝑧∈𝕌.

2. Coefficient Bounds for the Function Class 𝒢_Σ(𝛼,𝜆)

We begin by finding the estimates on the coefficients |𝑎₂| and |𝑎₃| for functions in the class 𝑓∈𝒢_Σ(𝛼,𝜆).

Theorem 4. Let 𝑓(𝑧) given by (1) be in the class 𝒢_Σ(𝛼,𝜆),  0<𝛼≤1, and 0≤𝜆<1. Then 𝑎₂≤2𝛼(1−𝜆)√,𝑎1+𝛼₃≤4𝛼²(1−𝜆)²+𝛼.1−𝜆 (7)(8)

Proof. It follows from (4) that 𝑧𝑓^′(𝑧)(1−𝜆)𝑓(𝑧)+𝜆𝑧𝑓^′(𝑧)=𝑝(𝑧)^𝛼,𝑤𝑔^′(𝑤)(1−𝜆)𝑔(𝑤)+𝜆𝑤𝑔^′(𝑤)=𝑞(𝑤)^𝛼, (9) where 𝑝(𝑧) and 𝑞(𝑤) in 𝒫 have the forms 𝑝(𝑧)=1+𝑝₁𝑧+𝑝₂𝑧²𝑞(𝑧)+⋯,=1+𝑞₁𝑤+𝑞₂𝑤²+⋯. (10)(11) Now, equating the coefficients in (9), we get (1−𝜆)𝑎₂=𝛼𝑝₁,𝜆²−1𝑎²₂+2(1−𝜆)𝑎₃=12𝛼(𝛼−1)𝑝²₁+2𝛼𝑝₂,−(1−𝜆)𝑎₂=𝛼𝑞₁,𝜆²−4𝜆+3𝑎²₂−2(1−𝜆)𝑎₃=12𝛼(𝛼−1)𝑞²₁+2𝛼𝑞₂. (12)(13)(14)(15) From (12) and (14), we get 𝑝₁=−𝑞₁,2(1−𝜆)²𝑎²₂=𝛼²𝑝²₁+𝑞²₁. (16)(17) From (13), (15), and (17), we obtain 𝑎²₂=𝛼²𝑝₂+𝑞₂(𝛼+1)(1−𝜆)². (18) Applying Lemma 3 for the coefficients 𝑝₂ and 𝑞₂, we immediately have 𝑎₂≤2𝛼(1−𝜆)().1+𝛼 (19) This gives the bound on |𝑎₂| as asserted in (7).
Next, in order to find the bound on |𝑎₃|, by subtracting (15) from (13), we get 4(1−𝜆)𝑎₃−4(1−𝜆)𝑎²₂=𝛼𝑝₂−𝑞₂+𝛼(𝛼−1)2𝑝²₁−𝑞²₁. (20) It follows from (16), (17), and (20) that 𝑎₃=𝛼𝑝₂−𝑞₂4()+𝛼1−𝜆²𝑝²₁+𝑞²₁2(1−𝜆)². (21) Applying Lemma 3 once again for the coefficients 𝑝₁,𝑝₂,𝑞₁,and 𝑞₂, we readily get 𝑎₃≤4𝛼²(1−𝜆)²+𝛼.1−𝜆 (22) This completes the proof of Theorem 4.

In the following section we find the estimates on the coefficients |𝑎₂| and |𝑎₃| for functions in the class ℳ_Σ(𝛽,𝜆).

3. Coefficient Bounds for the Function Class ℳ_Σ(𝛽,𝜆)

Theorem 5. Let 𝑓(𝑧) given by (1) be in the class ℳ_Σ(𝛽,𝜆),0≤𝛽<1, and 0≤𝜆<1. Then 𝑎₂≤21−𝛽,𝑎1−𝜆₃≤41−𝛽²(1−𝜆)²+1−𝛽.1−𝜆 (23)

Proof. It follows from (6) that there exists 𝑝,𝑞∈𝒫 such that 𝑧𝑓^′(𝑧)(1−𝜆)𝑓(𝑧)+𝜆𝑧𝑓^′(𝑧)=𝛽+1−𝛽𝑝(𝑧),𝑤𝑔^′(𝑤)(1−𝜆)𝑔(𝑤)+𝜆𝑤𝑔^′(𝑤)=𝛽+1−𝛽𝑞(𝑤), (24) where 𝑝(𝑧) and 𝑞(𝑤) have the forms of (10) and (11), respectively. Equating coefficients in (24) we get (1−𝜆)𝑎₂=1−𝛽𝑝₁,𝜆²−1𝑎²₂+2(1−𝜆)𝑎₃=1−𝛽𝑝₂,−(1−𝜆)𝑎₂=1−𝛽𝑞₁,𝜆²−4𝜆+3𝑎²₂−2(1−𝜆)𝑎₃=1−𝛽𝑞₂. (25)
The proof follows, by employing the techniques used in the proof of Theorem 4.

Taking 𝜆=0 in Theorems 4 and 5 one can get the following corollaries.

Corollary 6. Let 𝑓(𝑧) given by (1) be in the class 𝒮^𝛼_Σ and 0<𝛼≤1. Then 𝑎₂≤2𝛼√,𝑎𝛼+1³≤4𝛼²+𝛼. (26)

Corollary 7. Let 𝑓(𝑧) given by (1) be in the class 𝒮^∗_Σ(𝛽) and 0≤𝛽<1. Then 𝑎₂≤2−2𝛽,𝑎₃≤41−𝛽²+1−𝛽. (27)

Acknowledgment

The authors would like to record their sincere thanks to the referees for their valuable suggestions.