A refined Schwarz inequality on the boundary

Abstract

We consider a function f holomorphic in the unit disc D with f(D) ⊂ D and f(0) = f(z 0) = 0, for 0 < |z 0| < 1. We obtain sharp lower bounds on the angular derivative f′(c) at the point c where |c| = |f(c)| = 1.