Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

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


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30.42

Retrieve articles in all journals with MSC: 30.42


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0281901-6
PII: S 0002-9939(1971)0281901-6
Keywords: Modulus bounds, analytic functions, distortion, univalent, functions, Schwarzian derivative
Article copyright: © Copyright 1971 American Mathematical Society