Initial Coeﬃcient Estimates for New Families of m-Fold Symmetric Bi-univalent Functions

In the present work, we deﬁne two new families of analytic and m-fold symmetric biunivalent functions in the open unit disk ∆. Also, for functions in each of the classes introduced here, we prove upper bounds for the initial coeﬃcients | b m +1 | and | b 2 m +1 | . Furthermore, we get new special cases for our results.


Introduction
Let A be the class of analytic functions in the open unit disk ∆ = {t : t ∈ C, |t| < 1} and normalized by the conditions g(0) = 0 = g (0) − 1 and having the form shown below: Also, the class of all functions in A that are univalent in ∆ is denoted by S.
The Koebe one-quarter theorem [4] ensures that the image of ∆ under every univalent function g(t) ∈ A contains the disk of radius 1/4. Thus every univalent A function g ∈ A is said to be bi-univalent in ∆, if both g(t) and g −1 (t) are univalent in ∆. We denote by Σ the class of all bi-univalent functions in ∆ given by the Taylor-Maclaurin series expansion (1.1). Lewin [7] discussed the class of bi-univalent functions Σ and proved that the bound for the second coefficients of every g ∈ Σ satisfies the inequality |b 2 | ≤ 1.51. Motivated by the work of Lewin [7], Brannan and Clunie [3] hypothesised that |b 2 | ≤ √ 2. Some examples of bi-univalent functions are t 1−t , − log(1 − t) and 1 2 log 1+t 1−t (see also Srivastava et al. [15]). The coefficient estimate problem involving the bound of |b n | (n ∈ N\{1, 2}) for every g ∈ Σ is still an open problem [15].
For each function g ∈ S, the function is univalent and maps the unit disk ∆ into a region with m-fold symmetry. A function is told to be m-fold symmetric (see [5,10] ) if it has the following normalized form: We symbolize by S m the class of m-fold symmetric analytic univalent functions in ∆, which are normalized by the series expansion (1.4). In fact, the functions in the class S are one-fold symmetric (that is, m = 1).
Analogous to the concept of m-fold symmetric univalent functions, we here introduced the concept of m-fold symmetric analytic bi-univalent function. Each function g ∈ Σ generates an m-fold symmetric analytic bi-univalent function for each integer m ∈ N. The normalized form of g is given as in (1.4) and the series expansion for g −1 , which has been recently proven by Srivastava et al. [16], is given as follows: (1.5) We symbolize by Σ m the class of m-fold symmetric analytic bi-univalent functions in ∆. For m = 1, formulation (1.5) synchronizes with formulation (1.2) of the class Σ. Some examples of m-fold symmetric analytic bi-univalent functions are listed below [16]: , respectively. Recently, many authors investigated bounds for various subclasses of m-fold bi-univalent functions (see [1,2,6,11,12,13,14,17]).
The purpose of this work is to introduce two new subclasses of function class Σ m and derive estimates on initial coefficients |b m+1 | and |b 2m+1 | for functions in these new subclasses. Many related classes are also found out and connections to earlier known results are made. We have to remember the following lemma here so as to derive our basic results. Lemma 1.1. ( [4]) If p ∈ P, then |c n | ≤ 2 for each n ∈ N, where P is the family of all functions p, analytic in ∆, for which R(p(t)) > 0 where p(t) = 1 + c 1 t + c 2 t 2 + · · · (t ∈ ∆).

Coefficient Bounds for the Function Class
LH Σ m (τ, λ, δ; α) Definition 2.1. A function g(t) ∈ Σ m given by (1.4) is told to be in the class LH Σm (τ, λ, δ; α) if the following conditions are fulfilled: where the function h = g −1 is given by (1.5) and .
Proof. It follows from (2.1) and (2.2) that where the functions p(t) and q(w) are in P and have the following series representations: and It follows from (2.5) and (2.6) that From (2.9) and (2.11) we get From (2.10), (2.12) and (2.14), we find Stratifying Lemma (1.1) for coefficients p 2m and q 2m , we get The final inequality provides the estimation for |b m+1 | given in (2.3).
Next, in order to find the bound on |b 2m+1 |, by subtracting (2.12) from (2.10), we get
(3.14) Applying Lemma (1.1) for coefficients p 2m and q 2m , we readily get The final inequality provides the estimation for |b m+1 | given in (3.3).
Next, in order to find the bound on |b 2m+1 |, by subtracting (3.10) from (3.8), we get By substituting the value of b 2 m+1 from (3.12), we find Applying Lemma (1.1) once again for coefficients p 2m , p m , q 2m and q m , we easily obtain This proves Theorem (3.1).