On one filtration of holomorphic functions

TL;DR

This paper introduces a new family of analytic function classes based on the hypergeometric function, exploring their geometric properties, coefficient bounds, and their role in a filtration of infinitesimal generators linked to dynamical systems.

Contribution

It constructs a novel filtration of holomorphic functions using hypergeometric functions and analyzes their geometric and dynamical properties, connecting to Ma--Minda starlike functions.

Findings

Established sharp coefficient estimates for the classes.

Demonstrated the filtration of infinitesimal generators.

Linked the filtration to Ma--Minda starlike functions.

Abstract

In this work we consider a family of function classes constructed by means of the Gauss hypergeometric function $_2F_1(1,1;2;z) =-\frac{\log(1-z)}{z}$. We demonstrate that this family, in fact, constitutes classes of analytic functions subject to prescribed constraints on their derivatives. For these classes we obtain some geometric characteristics, including sharp coefficient estimates. Moreover, we show that this family naturally provides a filtration of infinitesimal generators, and investigate the corresponding dynamical behavior of the associated semigroups. It is interesting that this filtration links to the Ma--Minda starlike functions.