Coefficient estimates of analytic endomorphisms of the unit disk fixing a point with applications to concave functions

TL;DR

This paper investigates the coefficient regions of analytic self-maps of the unit disk fixing a point and applies these results to solve the Fekete-Szeg ext{"o} problem for normalized concave functions with a pole.

Contribution

It provides new coefficient estimates for analytic endomorphisms fixing a point and applies these to a classical extremal problem in geometric function theory.

Findings

Derived coefficient regions for fixed-point analytic self-maps

Solved the Fekete-Szeg ext{"o} problem for a class of concave functions

Established applications to extremal problems in geometric function theory

Abstract

In this note, we discuss the coefficient regions of analytic self-maps of the unit disk with a prescribed fixed point. As an application, we solve the Fekete-Szeg\H{o} problem for normalized concave functions with a prescribed pole in the unit disk.