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Branch point area methods in conformal mapping | | Haakan Hedenmalm
; Natalia Abuzyarova
; | | Date: |
17 Jun 2004 | | Subject: | Complex Variables MSC-class: 30C55 | math.CV | | Abstract: | The classical estimate of Bieberbach -- that $|a_2|le2$ for a given univalent function $phi(z)=z+a_2z^2+...$ in the class $S$ -- leads to best possible pointwise estimates of the ratio $phi’’(z)/phi’(z)$ for $phiin S$, first obtained by Koe{}be and Bieberbach. For the corresponding class $Sigma$ of univalent functions in the exterior disk, Goluzin found in 1943 -- by extremality methods -- the corresponding best possible pointwise estimates of $psi’’(z)/psi’(z)$ for $psiinSigma$. It was perhaps surprising that this time, the expressions involve elliptic integrals. Here, we obtain the area-type theorem which has Goluzin’s pointwise estimate as a corollary. This shows that the Koe{}be-Bieberbach estimate as well as that of Goluzin are both firmly rooted in the 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. | | Source: | arXiv, math.CV/0406347 | | Services: | Forum | Review | PDF | Favorites | |
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