On a theorem of Landau and Toeplitz

TL;DR

This paper revisits a 1907 theorem by Landau and Toeplitz related to the diameter of the image set, providing a streamlined proof and extending it to include growth bounds on the maximum modulus.

Contribution

It offers a simplified proof of Landau and Toeplitz's theorem and extends it to encompass bounds on the growth of the maximum modulus of functions.

Findings

Streamlined proof of Landau and Toeplitz's theorem.

Extension to bounds on the growth of the maximum modulus.

Clarification of the relation between image set diameter and function modulus.

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 streamlined proof of this result and also extend it to include bounds on the growth of the maximum modulus.