A topological generalization of Schwarz' lemma

The classical lemma of Schwarz states that if f(a) = zg(z) where g(z) is holomorphic inside the unit circle, and if |f(z)| < 1 when |s| < 1, then |f(z)| < |z| when |z| < 2. The proof for this is based on the fast that if f(s) is holomorphic in a region, its absolute value has no relative maxima in the interior of the region. Max Zorn has stated and proved a more general, highly axiomatic version of Schwarz' lemma, applicable to certaln families of transformations of a metrizable topological space into itself. But a geometrical interpretation of Zorn*s results is impossible without severe additional restrictions on the space and the families of transformations. The present paper proves a geometrical version of Schwarz' lemma by metrical-topological methods. The proof is applicable to the euclidean KL -space in particular, and, more generally, to any convex topological space provided each point of the space lies on one of the surfaces of a system of concentric compact spheres.

Name thesis:degree_name

Master of Arts

Level thesis:degree_level

Masters

Discipline thesis:degree_discipline

Natural Sciences

Grantor

Rice University

Year dc:date.issued

1941

Author dc:creator

• Piranian, George

Statement dc:rights

• Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1911/90181

OAI identifier oai:identifier

oai:repository.rice.edu:1911/90181