Lower Schwarz-Pick estimates and angular derivatives

Abstract

The well-known Schwarz-Pick lemma states that any analytic mapping $\phi$ of the unit disk $U$ into itself satisfies the inequality $$|\phi'(z)|\leq \frac{1-|\phi(z)|^2}{1-|z|^2}, \quad z\in U.$$ This estimate remains the same if we restrict ourselves to univalent mappings. The lower estimate is $|\phi'(z)|\geq 0$ generally or $|\phi'(z)|> 0$ for univalent functions. To make the lower estimate non-trivial we consider univalent functions and fix the angular limit and the angular derivative at some points of the unit circle. In order to obtain sharp estimates we make use of the reduced modulus of a digon.