A note on the squeezing function

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).