Title: 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 $D_1$ 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-González
that takes into account the geometry of $\partial{D_1}$. We also provide a new
estimate for the Kobayashi metric on bounded convex domains.