On Coefficient problem for bi-univalent analytic functions

TL;DR

This paper provides improved bounds for the initial coefficients of bi-univalent functions within specific classes, introducing new estimates for the fourth coefficient and bounds for the fifth in certain subclasses.

Contribution

It offers new coefficient bounds for bi-univalent functions, enhancing previous estimates and including novel bounds for the fourth and fifth coefficients in specific subclasses.

Findings

Improved bounds for second and third coefficients.

New estimate for the fourth coefficient.

Bound for the fifth coefficient in bi-starlike subclasses.

Abstract

Estimates for initial coefficients of Taylor-Maclaurin series of bi-univalent functions belonging to certain classes defined by subordination are obtained. Our estimates improve upon the earlier known estimates for second and third coefficient. The bound for the fourth coefficient is new. In addition, bound for the fifth coefficient is obtained for bi-starlike and strongly bi-starlike functions of order $\rho$ and $\beta$ respectively.