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

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 $\partial{D_1}$. We also provide a new estimate for the Kobayashi metric on bounded convex domains.