A lower bound for the K\"ahler-Einstein distance from the Diederich-Forn{\ae}ss index

In this note we establish a lower bound for the distance induced by the K"ahler-Einstein metric on pseudoconvex domains with positive hyperconvexity index (e.g. positive Diederich-Fornaess index). A key step is proving an analog of the Hopf lemma for Riemannian manifolds with Ricci curvature bounded from below.