{"id":{"repo_id":"rice","oai_identifier":"oai:repository.rice.edu:1911/90181"},"canonical_url":"https://search.dev.ndltd.org/etd/rice/oai:repository.rice.edu:1911/90181","repository":{"repo_id":"rice","name":"Rice University","base_url":"https://repository.rice.edu/server/oai/request"},"display":{"title":"A topological generalization of Schwarz' lemma","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Piranian, George"],"institution":"Rice University","degree_name":"Master of Arts","degree_level":"Masters","degree_discipline":"Natural Sciences","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1941,"date_issued":"1941","date_published":"1941","updated_at":"2026-07-24T04:10:19Z","subjects":[],"languages":["eng"],"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."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1911/90181","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Piranian, George"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2016-04-22T22:00:11Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2016-04-22T22:00:11Z"]},{"key":"dc:date.issued","label":"Date","values":["1941"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Natural Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Arts"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Rice University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["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."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1911/90181"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["A topological generalization of Schwarz' lemma"]}]}],"canonical_facts":{"dc:creator":["Piranian, George"],"dc:date.accessioned":["2016-04-22T22:00:11Z"],"dc:date.available":["2016-04-22T22:00:11Z"],"dc:date.issued":["1941"],"dc:description.abstract":["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."],"dc:identifier.uri":["https://hdl.handle.net/1911/90181"],"dc:language.iso":["eng"],"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."],"dc:title":["A topological generalization of Schwarz' lemma"],"dc:type":["Thesis"],"thesis:degree_discipline":["Natural Sciences"],"thesis:degree_level":["Masters"],"thesis:degree_name":["Master of Arts"],"thesis:institution_name":["Rice University"]},"updated_at":"2026-07-24T04:10:19Z"}