On the hypergeometric function and families of holomorphic functions

TL;DR

This paper explores a specific family of holomorphic functions in the unit disk, revealing their set-theoretic properties, the structure of filtrations, and introducing new concepts like quasi-infima, with surprising roles for hypergeometric functions.

Contribution

It provides a detailed set-theoretic analysis of a two-parameter family of holomorphic functions, introduces the concepts of quasi-infima and quasi-suprema, and uncovers new properties of the hypergeometric function.

Findings

The family is not a lattice.

Complete description of quasi-infima and quasi-suprema.

New properties of the hypergeometric function discovered.

Abstract

In this work, we examine one two-parameter family of sets consisting of functions holomorphic in the unit disk, previously investigated by several mathematicians. We focus on the set-theoretic properties of this family, identify the general form of filtrations within it, and discover that it is not a lattice. This insight motivates us to introduce a refined concept of quasi-infima and quasi-suprema, and to establish their complete description. Unexpectedly, some new properties of the Gau\ss\ hypergeometric function play a crucial role in our investigation.