The Bloch constant of bounded analytic functions on a multiply connected domain

Colonna, Flavia

doi:10.1090/S0002-9939-1991-1039529-9

The Bloch constant of bounded analytic functions on a multiply connected domain
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Proc. Amer. Math. Soc. 112 (1991), 1055-1066 Request permission

Abstract:

Let $F$ be an analytic function on the bounded domain $R$, a multiply-connected region of the complex plane. The Bloch constant of $F$ is defined by \[ \beta _F = \sup _{|z| < 1} (1 - |z|^2)|(F \circ p)’(z)|,\] where $p$ is a conformal universal cover of $R$ with domain $\Delta$, the open unit disk. If $F$ is bounded, then ${\beta _F} \leq ||F|{|_\infty }$, the sup-norm of $F$. In this paper we characterize those functions $F$ for which ${\beta _F} = ||F|{|_\infty }$ in terms of the zeros of $F$ when the boundary of $R$ is the union of finitely many curves. We conclude this paper by showing the existence of extremal functions, and generalizing the results to bounded harmonic mappings on these domains.

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