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Proceedings of the American Mathematical Society

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



A distortion theorem for analytic functions

Author: M. S. Robertson
Journal: Proc. Amer. Math. Soc. 28 (1971), 551-556
MSC: Primary 30.42
MathSciNet review: 0281901
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Abstract: Let $ f(z)$ be a function analytic in the disk $ E\{ z:\vert z\vert < 1\} $ and for some real number $ n > 0$ let $ \vert f(z)\vert \leqq {(1 - \vert z{\vert^2})^{ - n}},z \in E$. In this paper it is shown that

$\displaystyle \vert f'(z)\vert \leqq \frac{{{{(n + 1)}^{n + 1}}}}{{{n^n}}}\left... ...t^2})}^{2n}}\vert f(z){\vert^2}} \right] \div {(1 - \vert z{\vert^2})^{n + 1}},$

$ z \in E$. In the special case $ n = 1$ there is a constant $ K,3 \leqq K \leqq 4$, so that

$\displaystyle f'(z)\vert + \vert f(z){\vert^2} \leqq K{(1 - \vert z{\vert^2})^{ - 2}}.$

This result has application in univalent function theory.

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Keywords: Modulus bounds, analytic functions, distortion, univalent, functions, Schwarzian derivative
Article copyright: © Copyright 1971 American Mathematical Society