On some properties of bi-univalent functions in the unit disc

TL;DR

This paper investigates properties of bi-univalent functions in the unit disc, deriving bounds on initial coefficients, their differences, logarithmic coefficients, and Hankel determinants using Grunsky coefficients.

Contribution

It introduces a method based on Grunsky coefficients to establish bounds on various coefficients and determinants for bi-univalent functions.

Findings

Derived upper bounds for initial coefficients

Established bounds for the difference of consecutive coefficients

Provided bounds for the second Hankel determinant

Abstract

In this paper we use a method based on the Grunsky coefficients to find upper bounds of the modulus of the initial coefficients, difference of the moduli of two consecutive initial coefficients, of the modulus of the initial logarithmic coefficient, and of the second Hankel determinant for the class of normalized bi-univalent functions.