Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An area theorem for holomorphic functions
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by Shinji Yamashita
Proc. Amer. Math. Soc. 93 (1985), 615-617
DOI: https://doi.org/10.1090/S0002-9939-1985-0776189-3

Abstract:

Let $D \subsetneqq \mathbf {C}$ and $\Delta \subsetneqq \mathbf {C}$ be open sets in the plane $\mathbf {C}$ containing 0. Suppose that $D$ is connected and $f$ is holomorphic in $D$ with $f(0) = 0$. Then, for $Q = f(D) \cap \Delta$ and $P = {f^{ - 1}}(Q)$, we have \[ A(P) + A(Q) \geqslant \pi {[\min \{ \operatorname {dis}(\partial D,0),\operatorname {dis}(\partial \Delta ,0)\} ]^2},\] $A( \cdot )$ denoting the area. The constant $\pi$ is sharp and the equality holds if and only if $f \equiv 0,\operatorname {dis}(\partial D,0) \leqslant \operatorname {dis}(\partial \Delta ,0)$, and $D$ is a disk of center 0.
References
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Bibliographic Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 93 (1985), 615-617
  • MSC: Primary 30C55
  • DOI: https://doi.org/10.1090/S0002-9939-1985-0776189-3
  • MathSciNet review: 776189