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- W2949508347 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." @default.
- W2949508347 created "2019-06-27" @default.
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- W2949508347 date "2004-06-17" @default.
- W2949508347 modified "2023-09-27" @default.
- W2949508347 title "Branch point area methods in conformal mapping" @default.
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