A distortion theorem for analytic functions
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- by M. S. Robertson PDF
- Proc. Amer. Math. Soc. 28 (1971), 551-556 Request permission
Abstract:
Let $f(z)$ be a function analytic in the disk $E\{ z:|z| < 1\}$ and for some real number $n > 0$ let $|f(z)| \leqq {(1 - |z{|^2})^{ - n}},z \in E$. In this paper it is shown that \[ |f’(z)| \leqq \frac {{{{(n + 1)}^{n + 1}}}}{{{n^n}}}\left [ {1 - {{\left ( {\frac {n}{{n + 1}}} \right )}^{2n}}{{(1 - |z{|^2})}^{2n}}|f(z){|^2}} \right ] \div {(1 - |z{|^2})^{n + 1}},\] $z \in E$. In the special case $n = 1$ there is a constant $K,3 \leqq K \leqq 4$, so that \[ f’(z)| + |f(z){|^2} \leqq K{(1 - |z{|^2})^{ - 2}}.\] This result has application in univalent function theory.References
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Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 28 (1971), 551-556
- MSC: Primary 30.42
- DOI: https://doi.org/10.1090/S0002-9939-1971-0281901-6
- MathSciNet review: 0281901