Coefficient estimates for close-to-convex functions with argument $\beta$

TL;DR

This paper derives sharp bounds for coefficients of close-to-convex functions with argument beta, solving a problem posed by Goodman and Saff, and provides explicit estimates for the third coefficient using Herglotz representation.

Contribution

It introduces new coefficient bounds for close-to-convex functions with argument beta, including solving a previously open problem and computing explicit third coefficient estimates.

Findings

Sharp bounds for coefficients of close-to-convex functions obtained.

Solved the problem posed by Goodman and Saff.

Explicit estimate of the third coefficient provided.

Abstract

This paper deals with coefficient estimates for close-to-convex functions with argument $\beta$ ($-\pi/2<\beta<\pi/2$). By using Herglotz representation formula, sharp bounds of coefficients are obtained. In particluar, we solve the problem posed by A. W. Goodman and E. B. Saff in $\cite{GS}$. Finally some complicted computations yield the explicit estimate of the third coefficient.