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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Distortion theorems for Bloch functions


Authors: Xiang Yang Liu and David Minda
Journal: Trans. Amer. Math. Soc. 333 (1992), 325-338
MSC: Primary 30C75; Secondary 30C25, 30C80, 30D45
MathSciNet review: 1055809
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Abstract: We establish various distortion theorems for both normalized locally schlicht Bloch functions and normalized Bloch function with branch points. These distortion theorems give lower bounds on either $ \vert f\prime(z)\vert$ or $ \operatorname{Re} f\prime(z)$; most of our distortion theorems are sharp and all extremal functions identified. The main tools used in establishing these distortion theorems are the classical form of Julia's Lemma and a new version of Julia's Lemma that applies to certain multiple-valued analytic functions. As applications of these distortion theorems, we obtain known lower bounds for various Bloch constants and also establish improved lower bounds on a number of Marden constants for Bloch, normal and Yosida functions.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1055809-0
PII: S 0002-9947(1992)1055809-0
Article copyright: © Copyright 1992 American Mathematical Society