Faber Polynomial Coefficient Estimates for a Comprehensive Subclass of Analytic Bi-Univalent Functions Defined by Subordination

Abstract

In this paper, we find coefficient estimates by a new method making use of the Faber polynomial expansions for a comprehensive subclass of analytic bi-univalent functions, which is defined by subordinations in the open unit disk. The coefficient bounds presented in this paper would generalize and improve some recent works appeared in the literature.

Coeficient estimates for a subclass of meromorphic bi-univalent functions defined by subordination

“…But the interest on coefficient estimates of the meromorphic univalent functions keep on by many researchers, see for example, [18,19,25,26]. Several authors by using Faber polynomial expansions obtained coefficient estimates |a n | for classes meromorphic bi-univalent functions and bi-univalent functions, see for example [10,12,13,14,15,16,17,28,27]. First we recall some definitions and lemmas that used in this work.…”

Second Hankel determinant for a subclass of analytic bi-univalent functions defined by subordination

“…Lewin [18] investigated the class Σ of bi-univalent functions and showed that |a 2 | < 1.51 for all the functions belonging to Σ . Recently, many researchers have introduced and investigated several interesting subclasses of the bi-univalent function class Σ and they have found nonsharp estimates on the first two Taylor-Maclaurin coefficients |a 2 | and |a 3 | (see, for example, [1,9,16,17,23,25,26,29]).…”

“…The problem of estimating the coefficients |a n | with n ≥ 4 is presumably still an open problem. Using the Faber polynomial expansions, several authors obtained coefficient estimates of |a n | for the functions belonging in different subclasses of bi-univalent functions (see, for example, [10][11][12][13]30]). First, we will recall some definitions and lemmas that will be used in this work.…”

The Second Hankel Determinant Problem for a Class of Bi-Univalent Functions

“…Recently, several researches have focused on studying the class Σ, which consists of the bi-univalent functions, and acquired non-sharp estimates on the Taylor-Maclaurin coefficients | | and | |, e.g. [2][3][4][5][6][7][8]. The coefficient estimate issue for certain subfamilies of class of Taylor-Maclaurin coefficients | | for 4 is presumably still a concern.…”

“…The coefficient estimate issue for certain subfamilies of class of Taylor-Maclaurin coefficients | | for 4 is presumably still a concern. Either way, some researchers have investigated the Faber polynomial expansions to obtain the upper bounds for various subclasses of class [9][10][11][12][13][14].…”