Branch point area methods in conformal mapping
2006 (English)In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 99, 177-198 p.Article in journal (Refereed) Published
The classical estimate of Bieberbach that vertical bar a(2)vertical bar <= 2 for a given univalent function phi(z) = z + a(2)z(2) +... in the class S leads to the best possible pointwise estimates of the ratio phi''(z)/phi'(z) for phi is an element of S, first obtained by K oe be and Bieberbach. For the corresponding class E of univalent functions in the exterior disk, Goluzin found in 1943 by variational methods the corresponding best possible pointwise estimates of psi(z)/psi'(z) for psi is an element of Sigma. It was perhaps surprising that this time, the expressions involve elliptic integrals. Here, we obtain an area-type theorem which has Goluzin's pointwise estimate as a corollary. This shows that Goluzin's estimate, like the K oe be-Bieberbach estimate, is firmly rooted in area-based methods. The appearance of elliptic integrals finds a natural explanation: they arise because a certain associated covering surface of the Riemann sphere is a torus.
Place, publisher, year, edition, pages
2006. Vol. 99, 177-198 p.
IdentifiersURN: urn:nbn:se:kth:diva-16253ISI: 000243386000005ScopusID: 2-s2.0-33847712912OAI: oai:DiVA.org:kth-16253DiVA: diva2:334295
QC 201005252010-08-052010-08-05Bibliographically approved