An area theorem for holomorphic functions
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- by Shinji Yamashita PDF
- Proc. Amer. Math. Soc. 93 (1985), 615-617 Request permission
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
- H. Alexander, B. A. Taylor, and J. L. Ullman, Areas of projections of analytic sets, Invent. Math. 16 (1972), 335–341. MR 302935, DOI 10.1007/BF01425717
- H. Alexander and R. Osserman, Area bounds for various classes of surfaces, Amer. J. Math. 97 (1975), no. 3, 753–769. MR 380596, DOI 10.2307/2373775
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655 O. Frostman, Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions, Medd. Lunds Univ. Mat. Sem. 3 (1935), 1-118.
- J. L. Ullman, An area theorem for schlicht functions, Amer. Math. Monthly 80 (1973), 184–186. MR 315118, DOI 10.2307/2318377
Additional 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