Maclaurin Coeﬃcient Estimates for a New General Subclasses of m-Fold Symmetric Holomorphic Bi-Univalent Functions

The purpose of the present paper is to introduce and investigate two new general subclasses MA Σ m ( δ, λ ; α ) and MA Σ m ( δ, λ ; β ) of Σ m consisting of holomorphic and m-fold symmetric bi-univalent functions deﬁned in the open unit disk U . For functions belonging to the two classes introduced here, we derive estimates on the initial coeﬃcients | d m +1 | and | d 2 m +1 | . We get new special cases for our results. In addition, Several related classes are also investigated and connections to earlier known outcomes are made


Introduction
Let A denote the class of functions of the form: which are holomorphic in the open unit disk U = {s : |s| < 1}, and let S be the subclass of A consisting of the form (1.1) which are also univalent in U .The A function k ∈ A is said to be bi-univalent in U , if both k(s) and k −1 (s) are univalent in U .We denote by Σ the class of all bi-univalent functions in U 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 k ∈ 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 s 1−s , − log(1 − s) and 1 2 log 1+s 1−s (see also Srivastava et al. [13]).The coefficient estimate problem involving the bound of |d n | (n ∈ N\{1, 2}) for every k ∈ Σ is still an open problem [13].
For each function k ∈ S, the function is univalent and maps the unit disk U into a region with m-fold symmetry.
A function is said to be m-fold symmetric (see [5], [9]) if it has the following normalized form: We denote by S m the class of m-fold symmetric univalent functions in U , which are normalized by the series expansion (1.4).In fact, the functions in the class S are one-fold symmetric.Analogous to the concept of m-fold symmetric univalent functions, we here introduced the concept of m-fold symmetric bi-univalent functions.Each function k ∈ Σ generates an m-fold symmetric bi-univalent function for each integer m ∈ N. The normalized form of k is given as in (1.4) and the series expansion for k −1 , which has been recently proven by Srivastava et al. [14], is given as follows: where k −1 = h.We denote by Σ m the class of m-fold symmetric bi-univalent functions in U .For m = 1, formula (1.5) coincides with formula (1.2) of the class Σ.
The aim of this paper is to derive estimates on the initial coefficients |d m+1 | and |d 2m+1 | for functions belonging to the new general subclasses MA Σm (δ, λ; α) and MA Σm (δ, λ; β) of Σ m .We get new special cases for our results.In addition, Several related classes are also investigated and connections to earlier known outcomes are made.

2(2m
Or equivalently By substituting the value of d 2 m+1 from (3.12), we find .17) By using Lemma (1.1) once again for the coefficients p m , p 2m , q m and q 2m we get .