Generalizing Certain Analytic Functions Correlative to the n-th Coefficient of Certain Class of Bi-Univalent Functions

Hameed Ur Rehman,Maslina Darus,Jamal Salah,Efthymios G Tsionas

doi:10.1155/2021/6621315

Hameed Ur Rehman, Maslina Darus + Show 2 more

Open Access

https://doi.org/10.1155/2021/6621315

Copy DOI

Abstract

In the present paper, the authors implement the two analytic functions with its positive real part in the open unit disk. New types of polynomials are introduced, and by using these polynomials with the Faber polynomial expansion, a formula is structured to solve certain coefficient problems. This formula is applied to a certain class of bi-univalent functions and solve the n -th term of its coefficient problems. In the last section of the article, several well-known classes are also extended to its n -th term.

Highlights

The structural properties and information about GeometricFunction Theory depends on the estimation of coefficient of analytic functions
Srivastava et al [14] have made the use of Faber polynomial expansions with q-analysis to determine the bounds for the n-th coefficient in the Taylor–Maclaurin series expansions
Srivastava et al [15] made the use of a linear combination of three functions ((f(z)/z), f′ (z), and zf′′ (z)) with the technique involving the Faber polynomials and determined the coefficient estimates for the general Taylor–Maclaurin functions belonging to the bi-univalent function

Summary

Introduction

Function Theory depends on the estimation of coefficient of analytic functions. For example, the second coefficient estimation (|a2 |) in the set of univalent functions gives the growth, distortion bounds, and covering theorems. Taha [2] conjectured the coefficient bounds for the classes of bi-univalent functions. Srivastava et al [14] have made the use of Faber polynomial expansions with q-analysis to determine the bounds for the n-th coefficient in the Taylor–Maclaurin series expansions. Srivastava et al [15] made the use of a linear combination of three functions ((f(z)/z), f′ (z), and zf′′ (z)) with the technique involving the Faber polynomials and determined the coefficient estimates for the general Taylor–Maclaurin functions belonging to the bi-univalent function. Let A denotes the class of analytic functions in the unit disk, {U z ∈ C: |z| < 1} of the following form:. Köebe’s theorem ensures that the image of a unit disc U, under every univalent function f ∈ T, contains a disk of radius (1/4). The series expansion of the inverse of f ∈ T in some disk about the origin is given by f− 1 (w) w + a2 w2 + a3 w3 + a4 w4 + a5 w5 + · · ·

Discussion

Main Results

Analytic Functions Correlative to Bi-Univalent Functions

Corollaries and Consequences

Conclusions