A form of Schwarz's lemma and a bound for the Kobayashi metric on convex domains

TL;DR

This paper introduces a new form of Schwarz's lemma for convex domains and derives bounds for the Kobayashi metric, enhancing understanding of holomorphic maps' behavior in convex complex analysis.

Contribution

It presents a novel Schwarz's lemma for convex domains and establishes improved bounds for the Kobayashi metric, considering boundary geometry.

Findings

Lower bounds on images of compact sets under holomorphic maps

Enhanced estimates for the Kobayashi metric on convex domains

Incorporation of boundary geometry into metric bounds

Abstract

We present a form of Schwarz's lemma for holomorphic maps between convex domains $D_1$ and $D_2$. This result provides a lower bound on the distance between the images of relatively compact subsets of $D_1$ and the boundary of $D_2$. This is a natural improvement of an old estimate by Bernal-Gonz\'alez that takes into account the geometry of $\partial{D_1}$. Using similar techniques, we also provide a new estimate for the Kobayashi metric on bounded convex domains.