The Fekete--Szeg\"{o} problem and filtration of generators

TL;DR

This paper explores the Fekete--Szeg"{o} functional within the context of filtration classes of infinitesimal generators, introducing new classes via a nonlinear differential operator and establishing bounds for the functional.

Contribution

It introduces new filtration classes of infinitesimal generators using a nonlinear differential operator and determines sharp bounds for the Fekete--Szeg"{o} functional on these classes.

Findings

Established properties of new filtration classes.

Derived sharp upper bounds for the Fekete--Szeg"{o} functional.

Presented open problems for future research.

Abstract

In this paper we study an interpolation problem involving the Fekete--Szeg\"{o} functional. It turns out that this problem links to the so-called filtration of infinitesimal generators. We introduce new filtration classes using the non-linear differential operator \[\alpha\frac{f(z)}{z}+\beta\frac{zf'(z)}{f(z)}+(1-\alpha-\beta)\left(1+\frac{zf''(z)}{f'(z)}\right)\] and establish certain properties of these classes. Sharp upper bounds of the modulus of the Fekete--Szeg\"{o} functional over some filtration classes are found. We also present open problems for further study.