Initial Coefficients of Bi-univalent Functions

TL;DR

This paper investigates bounds on the initial coefficients of bi-univalent functions, which are analytic and univalent in the unit disk, by exploring subclasses of such functions and generalizing previous results.

Contribution

It provides new coefficient estimates for bi-univalent functions belonging to specific subclasses, extending and unifying earlier findings in the field.

Findings

Derived bounds for initial coefficients of bi-univalent functions

Unified previous results as special cases

Enhanced understanding of bi-univalent function subclasses

Abstract

An analytic function $f$ defined on the open unit disk $\mathbb{D}=\{z:|z|<1\}$ is bi-univalent if the function $f$ and its inverse $f^{-1}$ are univalent in $\mathbb{D}$. Estimates for the initial coefficients of bi-univalent functions $f$ are investigated when $f$ and $f^{-1}$ respectively belong to some subclasses of univalent functions. Some earlier results are shown to be special cases of our results.