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Length and area estimates of the derivatives of bounded holomorphic functions


Author: Shinji Yamashita
Journal: Proc. Amer. Math. Soc. 88 (1983), 29-33
MSC: Primary 30C50
DOI: https://doi.org/10.1090/S0002-9939-1983-0691273-9
MathSciNet review: 691273
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Abstract: MacGregor [1] and Yamashita [5] proved the estimates of the coefficient $ {a_n}$ of the Taylor expansion $ f(z) = {a_0} + {a_n}{z^n} + \cdots $ of $ f$ nonconstant and holomorphic in $ \vert z\vert < 1$ in terms of the area of the image of $ \vert z\vert < r < 1$ by $ f$ and the length of its outer or exact outer boundary. We shall consider some analogous estimates in terms of the non-Euclidean geometry for $ f$ bounded, $ \vert f\vert < 1$, in $ \vert z\vert < 1$. For example, $ 2\pi {r^n}\vert{a_n}\vert/(1 - \vert{a_0}{\vert^2})$ is strictly less than the non-Euclidean length of the boundary of the image of $ \vert z\vert < r$, the multiplicity not being counted.


References [Enhancements On Off] (What's this?)

  • [1] T. H. MacGregor, Length and area estimates for analytic functions, Michigan Math. J. 11 (1964), 317-320. MR 0171003 (30:1236)
  • [2] -, Translations of the image domains of analytic functions, Proc. Amer. Math. Soc. 16 (1965), 1280-1286. MR 0194600 (33:2810)
  • [3] W. K. Hayman, Mullivalent functions, Cambridge Univ. Press, London, 1967.
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  • [5] S. Yamashita, Length estimates for holomorphic functions, Proc. Amer. Math. Soc. 81 (1981), 250-252. MR 593467 (83c:30018)

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DOI: https://doi.org/10.1090/S0002-9939-1983-0691273-9
Article copyright: © Copyright 1983 American Mathematical Society

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