The Sharp Inequalities on the Logarithmic Coefficients for Complex-Valued Functions Linked to the Domain of a Four-Leaf Type Function

Abstract

A primary objective of examining coefficient inequalities for several families of complex-valued functions is to express their coefficients with respect to those of Carathéodory functions. Utilizing the inequalities related to Carathéodory functions enables effective analysis of the properties of coefficient inequalities. This article focuses on deriving sharp inequalities in the logarithmic coefficients for complex-valued functions related to a four-leaf-type function. The coefficient inequalities cover the four logarithmic coefficients, the Zalcman, Fekete-Szegö, and Krushkal inequalities for the specified family. Additionally, we present sharp estimates of second Hankel determinants, where the entries involve logarithmic coefficients. All the derived estimations are sharp.