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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An area theorem for holomorphic functions


Author: Shinji Yamashita
Journal: Proc. Amer. Math. Soc. 93 (1985), 615-617
MSC: Primary 30C55
MathSciNet review: 776189
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Abstract: Let $ D\mathop \subset \limits_ \ne {\mathbf{C}}$ and $ \Delta \mathop \subset \limits_ \ne {\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

$\displaystyle A(P) + A(Q) \geqslant \pi {[\min \{ \operatorname{dis}(\partial D,0),\operatorname{dis}(\partial \Delta ,0)\} ]^2},$

$ 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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0776189-3
PII: S 0002-9939(1985)0776189-3
Keywords: Areas of the image and the preimage, Alexander-Osserman-Taylor-Ullman's inequality, inner functions, Hardy class $ {H^2}$, Frostman's theorem
Article copyright: © Copyright 1985 American Mathematical Society