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On the Lower Order of Locally Univalent Functions

On the Lower Order of Locally Univalent Functions Let f be analytic and f′(z) ≠ 0 in D and let $A_{f}(z)={1-\mid z \mid^{2}\over 2}{f^{\prime \prime}(z)\over f\prime(z)}-{\overline z}\} {\rm for}\ z \ \epsilon\ D $ Many properties of f can be described by the (linear-invariant) order ${\rm sup}\mid A_{f}(z)\mid\atop \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!z\in {\rm D}$ The work of Avkhadiev and Wirths led to the introduction of the lower order of f defined by infz∈D ¦A f(z)¦. It is perhaps a surprise that there are many (necessarily unbounded) functions of positive lower order. This paper studies some properties of these functions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

On the Lower Order of Locally Univalent Functions

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Publisher
Springer Journals
Copyright
Copyright © 2008 by Heldermann  Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03321694
Publisher site
See Article on Publisher Site

Abstract

Let f be analytic and f′(z) ≠ 0 in D and let $A_{f}(z)={1-\mid z \mid^{2}\over 2}{f^{\prime \prime}(z)\over f\prime(z)}-{\overline z}\} {\rm for}\ z \ \epsilon\ D $ Many properties of f can be described by the (linear-invariant) order ${\rm sup}\mid A_{f}(z)\mid\atop \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!z\in {\rm D}$ The work of Avkhadiev and Wirths led to the introduction of the lower order of f defined by infz∈D ¦A f(z)¦. It is perhaps a surprise that there are many (necessarily unbounded) functions of positive lower order. This paper studies some properties of these functions.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Sep 20, 2007

References