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

Abstract

We present a form of Schwarz's lemma for holomorphic maps between convex domains D1 and D2. This result provides a lower bound on the distance between the images of relatively compact subsets of D1 and the boundary of D2. This is a natural improvement of an old estimate by Bernal-González that takes into account the geometry of ∂D1. Using similar techniques, we also provide a new estimate for the Kobayashi metric on bounded convex domains.