Distortion theorems for Bloch functions

Liu, Xiang Yang; Minda, David

doi:10.1090/S0002-9947-1992-1055809-0

Distortion theorems for Bloch functions
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by Xiang Yang Liu and David Minda PDF

Trans. Amer. Math. Soc. 333 (1992), 325-338 Request permission

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 $|f\prime (z)|$ 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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