Multi-point variations of the Schwarz lemma with diameter and width conditions

Abstract:

Suppose that $f$ is holomorphic in the unit disk $\mathbb D$ and $f(\mathbb D)\subset \mathbb D$, $f(0)=0$. A classical inequality due to Littlewood generalizes the Schwarz lemma and asserts that for every $w\in f(\mathbb D)$, we have $|w|\leq \prod _j |z_j(w)|$, where $z_j(w)$ is the sequence of pre-images of $w$. We prove a similar inequality by replacing the assumption $f(\mathbb D)\subset \mathbb D$ with the weaker assumption Diam$f(\mathbb D)=2$. This inequality generalizes a growth bound involving only one pre-image, proven recently by Burckel et al. We also prove growth bounds for holomorphic $f$ mapping $\mathbb D$ onto a region having fixed horizontal width. We give a complete characterization of the equality cases. The main tools in the proofs are the Green function and the Steiner symmetrization.