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Proceedings of the American Mathematical Society

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

 

 

The derivative of a bounded holomorphic function in the disk


Author: Shinji Yamashita
Journal: Proc. Amer. Math. Soc. 53 (1975), 60-64
MSC: Primary 30A72
DOI: https://doi.org/10.1090/S0002-9939-1975-0377061-7
MathSciNet review: 0377061
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Abstract: Let a nonconstant function $ f$ be holomorphic and bounded, $ \vert f\vert < 1$ in $ D:\vert z\vert < 1$. We shall estimate $ {f^{\ast}}(z) = (1 - \vert z{\vert^2})\vert f'(z)\vert/(1 - \vert f(z){\vert^2})$ at each point $ z\epsilon D$ ((1) in Theorem 1). The function $ d$ appearing in the estimate concerns the sizes of the schlicht disks on the Riemannian image $ \mathcal{F}$ of $ D$ by $ f$. Boundary properties of $ f$ and $ {f^{\ast}}$ will be stated in Theorems 2 and 3; use is made of the cluster sets of $ d$.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0377061-7
Keywords: Riemannian image, schlicht disks, non-Euclidean distance, the lemma of Schwarz and Pick, cluster sets
Article copyright: © Copyright 1975 American Mathematical Society