Families of inverse functions: coefficient bodies and the Fekete--Szeg\"{o} problem

TL;DR

This paper characterizes the coefficient bodies of inverse function families, identifies boundary functions, and derives sharp bounds for Fekete--Szeg"{o} functionals, also introducing a new Bell polynomial formula.

Contribution

It provides a comprehensive description of coefficient bodies for inverse functions and applies these results to obtain sharp bounds for Fekete--Szeg"{o} functionals, including a novel Bell polynomial formula.

Findings

Coefficient bodies for inverse functions are characterized.

Boundary functions for these bodies are explicitly described.

Sharp bounds for Fekete--Szeg"{o} functionals are established.

Abstract

In this paper we establish the coefficient bodies for a wide class of families of inverse functions. We also completely describe those functions that provide boundary points of that bodies in small dimensions. As an application we get sharp bounds for Fekete--Szeg\"{o} functionals over some classes of functions defined by quasi-subordination as well as over classes of their inverses. As a biproduct we derive a formula for ordinary Bell polynomials that seems to be new.