We introduce a unified subclass of the function class Σ of
biunivalent functions defined in the open unit disc. Furthermore, we find estimates on
the coefficients |a2| and |a3| for functions in this subclass. In addition, many relevant
connections with known or new results are pointed out.

1. Introduction

Let 𝒜 denote the class of functions of the form
(1)f(z)=z+∑n=2∞anzn
which are analytic in the open unit disc 𝕌={z:z∈ℂand|z|<1}. Further, by 𝒮, we will denote the class of all functions in 𝒜 which are univalent in 𝕌.

Some of the important and well-investigated subclasses of the univalent function class 𝒮 include, for example, the class 𝒮*(α) of starlike functions of order α in 𝕌 and the class 𝒦(α) of convex functions of order α in 𝕌.

It is well known that every function f∈𝒮 has an inverse f-1, defined by
(2)f-1(f(z))=z(z∈𝕌),f(f-1(w))=w(|w|<r0(f); r0(f)≥14),
where
(3)f-1(w)=w-a2w2+(2a22-a3)w3-(5a23-5a2a3+a4)w4+⋯.

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

In 1967, Lewin [1] investigated the biunivalent function class Σ and showed that |a2|<1.51; on the other hand Brannan and Clunie [2] (see also [3–5]) and Netanyahu [6] made an attempt to introduce various subclasses of biunivalent function class Σ and obtained nonsharp coefficient estimates on the first two coefficients |a2| and |a3| of (1). But the coefficient problem for each of the following Taylor-Maclaurin coefficients |an| for n∈ℕ∖{1,2}; ℕ:={1,2,3,…} is still an open problem. In this line, following Brannan and Taha [4], recently, many researchers have introduced and investigated several interesting subclasses of biunivalent function class Σ and they have found nonsharp estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3|; for details, one can refer to the works of [7–13].

Now, we define ℛΣ(α,λ) of function f∈𝒜 satisfying the following conditions:
(4)f∈Σ,|arg(z1-λf′(z)(f(z))1-λ)|<απ2,|arg(w1-λg′(w)(g(w))1-λ)|<απ2(z,w∈𝕌; λ≥0)
for some α(0<α≤1), where g(w) is the extension of f-1(w) to 𝕌. Similarly, we say that a function f∈𝒜 belongs to the class ℛΣ(β,λ) if f(z) satisfies the following inequalities:
(5)f∈Σ,ℜ(z1-λf′(z)(f(z))1-λ)>β,ℜ(w1-λg′(w)(g(w))1-λ)>β(z,w∈𝕌; λ≥0),
for some β(0≤β<1), where g(w) is the extension of f-1(w) to 𝕌. The classes ℛΣ(α,λ) and ℛΣ(β,λ) were introduced by Prema and Keerthi [14]; furthermore, for these classes, they have found the following estimates on the first two Taylor-Maclaurin coefficients in (1).

Theorem 1.

If f∈ℛΣ(α,λ), 0<α≤1, and λ≥0, then
(6)|a2|≤2α(α+1+λ)(1+λ),|a3|≤4α2(1+λ)2+2α2+λ.

Theorem 2.

If f∈ℛΣ(β,λ), 0≤β<1, and λ≥0, then
(7)|a2|≤2(1-β)1+λ,|a3|≤4(1-β)2(1+λ)2+2(1-β)2+λ.

Motivated by the works of Xu et al. [12, 13], we introduce the following generalized subclass ℛΣ(φ,ψ,λ) of the analytic function class 𝒜.

Definition 3.

Let f∈𝒜, and let the functions φ,ψ:𝕌→ℂ be so constrained that
(8)min{ℜ(φ(z)),ℜ(ψ(z))}>0(z∈𝕌),φ(0)=ψ(0)=1.
We say that f∈ℛΣ(φ,ψ,λ) if the following conditions are satisfied:
(9)f∈Σ,z1-λf′(z)(f(z))1-λ∈φ(𝕌),w1-λg′(w)(g(w))1-λ∈ψ(𝕌)(z,w∈𝕌),
where λ≥0 and the function g(w) is the extension of f-1(w) to 𝕌.

We note that by specializing λ, φ, and ψ, we get the following interesting subclasses:

ℛΣ(φ,ψ,1)=ℋΣφ,ψ; see [12],

ℛΣ(((1+z)/(1-z))α, ((1+z)/(1-z))α,λ)=ℛΣ(α,λ) (0<α≤1; λ≥0) and ℛΣ((1+(1-2β)z)/(1-z), (1+(1-2β)z)/(1-z),λ)=ℛΣ(β,λ) (0≤β<1; λ≥0); see [14],

ℛΣ(((1+z)/(1-z))α,((1+z)/(1-z))α,1)=ℋΣα (0<α≤1) and ℛΣ((1+(1-2β)z)/(1-z),(1+(1-2β)z)/(1-z),1)=ℋΣβ (0≤β<1); see [11].

The objective of the present paper is to introduce a new subclass ℛΣ(φ,ψ,λ) and to obtain the estimates on the coefficients |a2| and |a3| for the functions in theaforementioned class, employing the techniques used earlier by Xu et al. [12, 13].

2. Main Result

In this section, we find the estimates on the coefficients |a2| and |a3| for the functions in the class ℛΣ(φ,ψ,λ).

Theorem 4.

Let f(z) be of the form (1). If f∈ℛΣ(φ,ψ,λ), then
(10)|a2|≤|φ′′(0)|+|ψ′′(0)|8+4λ,(11)|a3|≤|φ′′(0)|4+2λ.

Proof.

Since f∈ℛΣ(φ,ψ,λ), from (9), we have,
(12)z1-λf′(z)(f(z))1-λ=φ(z)(z∈𝕌),w1-λg′(w)(g(w))1-λ=ψ(w)(w∈𝕌),
where
(13)φ(z)=1+φ1z+φ2z2+⋯,ψ(z)=1+ψ1z+ψ2z2+⋯
satisfy the conditions of Definition 3. Now, equating the coefficients in (12), we get
(14)(1+λ)a2=φ1,(15)(2+λ)a3=φ2,(16)-(1+λ)a2=ψ1,(17)(2+λ)(2a22-a3)=ψ2.
From (14) and (16), we get
(18)φ1=-ψ1,2(1+λ)2a22=φ12+ψ12.
From (15) and (17), we obtain
(19)a22=φ2+ψ22(2+λ).
Since φ(z)∈φ(𝕌) and ψ(z)∈ψ(𝕌), we immediately have
(20)|a2|≤|φ′′(0)|+|ψ′′(0)|8+4λ.
This gives the bound on |a2| as asserted in (10).

Next, in order to find the bound on |a3|, by subtracting (17) from (15), we get
(21)2(2+λ)a3-2(2+λ)a22=φ2-ψ2.
It follows from (19) and (21) that
(22)a3=φ22+λ.
Since φ(z)∈φ(𝕌) and ψ(z)∈ψ(𝕌), we readily get |a3|≤|φ′′(0)|/(4+2λ) as asserted in (11). This completes the proof of Theorem 4.

By setting φ(z)=ψ(z)=((1+Az)/(1+Bz))α, where -1≤B<A≤1 and 0<α≤1, in Theorem 4, we get the following corollary.

Corollary 5.

Let f(z) be of the form (1) and in the class ℛΣ(A,B,α,λ). Then,
(23)|a2|≤α2(A-B)2-α(A2-B2)4+2λ,|a3|≤α2(A-B)2-α(A2-B2)4+2λ.

If we choose A=1 and B=-1 in Corollary 5, we have the following corollary.

Corollary 6.

Let f(z) be of the form (1) and in the class ℛΣ(α,λ), 0<α≤1 and λ≥0. Then,
(24)|a2|≤α22+λ,|a3|≤2α22+λ.

Remark 7.

The estimates found in Corollary 6 would improve the estimates obtained in [14, Theorem 2.2].

If we set A=1-2β, B=-1, where 0≤β<1 and α=1 in Corollary 5, we readily have the following corollary.

Corollary 8.

Let f(z) be of the form (1) and in the class ℛΣ(β,λ), 0≤β<1 and λ≥0. Then
(25)|a2|≤2(1-β)2+λ,|a3|≤2(1-β)2+λ.

Remark 9.

The estimates found in Corollary 8 would improve the estimates obtained in [14, Theorem 3.2].

Remark 10.

For λ=1, the bounds obtained in Theorem 4 are coincident with the outcome of Xu et al. [12]. Taking λ=0 in Corollaries 6 and 8, the estimates on the coefficients |a2| and |a3|, are the improvement of the estimates on the first two Taylorû Maclaurin coefficients obtained in [10, Corollaries 2.3 and 3.3]. Also, for the choices of λ=1, the results stated in Corollaries 6 and 8 would improve the bounds stated in [11, Theorems 1 and 2], respectively. Furthermore, various other interesting corollaries and consequences of our main result could be derived similarly by specializing φ and ψ.

Acknowledgment

The authors would like to thank the referee for his valuable suggestions.

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