Area, capacity and diameter versions of Schwarz's Lemma

TL;DR

This paper explores geometric variants of Schwarz's Lemma, providing new proofs and extensions that relate the size of the image set to various geometric measures like diameter, capacity, and area.

Contribution

It introduces a unified approach to estimate the image size of holomorphic functions using multiple geometric quantities, extending classical results.

Findings

New proof of Landau and Toeplitz's diameter result

Extensions to bounds involving growth of maximum modulus

Development of a general geometric estimation framework

Abstract

The now canonical proof of Schwarz's Lemma appeared in a 1907 paper of Carath\'eodory, who attributed it to Erhard Schmidt. Since then, Schwarz's Lemma has acquired considerable fame, with multiple extensions and generalizations. Much less known is that, in the same year 1907, Landau and Toeplitz obtained a similar result where the diameter of the image set takes over the role of the maximum modulus of the function. We give a new proof of this result and extend it to include bounds on the growth of the maximum modulus. We also develop a more general approach in which the size of the image is estimated in several geometric ways via notions of radius, diameter, perimeter, area, capacity, etc...