2006/03/24 by Robert B. Burckel, Burckel, Robert B., Donald E. Marshall +3
Mathematics · #30C80 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #History and Overview (math.HO) #Holomorphic and Operator Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.math/0603579
openalex publication_date 2006/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The now canonical proof of Schwarz's Lemma appeared in a 1907 paper of Carathéodory, 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.