A note on the squeezing function

TL;DR

This paper discusses recent advances and new results on the squeezing function problem in complex analysis, focusing on how far boundary points of multiply connected domains can be mapped within the unit disk while fixing the origin.

Contribution

It reviews recent findings by Ng, Tang, Tsai, Gumenyuk, and Roth, and introduces new results using a method from previous research to analyze the squeezing function.

Findings

Recent results on boundary push limits in the squeezing problem

New bounds and properties of the squeezing function for multiply connected domains

Application of a novel method to derive new squeezing function estimates

Abstract

The squeezing problem on $\mathbb{C}$ can be stated as follows. Suppose that $\Omega$ is a multiply connected domain in the unit disk $\mathbb{D}$ containing the origin $z=0$. How far can the boundary of $\Omega$ be pushed from the origin by an injective holomorphic function $f:\Omega\to \mathbb{D}$ keeping the origin fixed? In this note, we discuss recent results on this problem obtained by Ng, Tang and Tsai (Math. Anal. 2020) and by Gumenyuk and Roth (arXiv:2011.13734, 2020) and also prove few new results using a method suggested in one of our previous papers (Zapiski Nauchn. Sem. POMI 1993).