Faber polynomial coefficient estimates for analytic bi-Bazilevič functions

Abstract

A function is said to be bi-univalent in the openunit disk $\mathbb{D}$ if both the function and its inverse areunivalent in $\mathbb{D}$. By the same token, a function is said tobe bi-Bazilevič in $\mathbb{D}$ if both the function and itsinverse are Bazilevič there. The behavior of these types offunctions are unpredictable and not much is known about theircoefficients. In this paper we use the Faber polynomial expansionsto find upper bounds for the coefficients of classes ofbi-Bazilevič functions. The coefficients bounds presented inthis paper are better than those so far appeared in theliterature. The technique used in this paper is also new and wehope that this will trigger further interest in applying ourapproach to other related problems.