Lower Schwarz-Pick estimates and angular derivatives

TL;DR

This paper investigates lower bounds for the angular derivatives of univalent functions in the unit disk, extending classical Schwarz-Pick estimates by using the reduced modulus of a digon to obtain sharp bounds.

Contribution

It introduces sharp lower estimates for angular derivatives of univalent functions at boundary points using the reduced modulus of a digon, extending classical Schwarz-Pick results.

Findings

Derived sharp lower bounds for angular derivatives at boundary points.

Extended classical Schwarz-Pick estimates to include lower bounds.

Applied the reduced modulus of a digon to obtain these estimates.

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.