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On Keogh’s Length Estimate for Bounded Starlike Functions

On Keogh’s Length Estimate for Bounded Starlike Functions For a bounded starlike function ƒ on the unit disc, we consider L(r), the length of the image of the circle ¦z¦ = r. Keogh showed that L(r) = O(log 1/(1 − r) as r → 1 and Hayman showed that this is the correct asymptotic. We give an alternative geometric construction which strengthens Hayman’s result, showing that the constant in Keogh’s original inequality is sharp. The analysis uses standard estimates on the hyperbolic metric of plane domains. The self-similarity of the construction allows for the examples to be expressed analytically. For context, we give a brief survey of related estimates on integral means and coefficients of univalent functions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

On Keogh’s Length Estimate for Bounded Starlike Functions

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Publisher
Springer Journals
Copyright
Copyright © 2005 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/BF03321098
Publisher site
See Article on Publisher Site

Abstract

For a bounded starlike function ƒ on the unit disc, we consider L(r), the length of the image of the circle ¦z¦ = r. Keogh showed that L(r) = O(log 1/(1 − r) as r → 1 and Hayman showed that this is the correct asymptotic. We give an alternative geometric construction which strengthens Hayman’s result, showing that the constant in Keogh’s original inequality is sharp. The analysis uses standard estimates on the hyperbolic metric of plane domains. The self-similarity of the construction allows for the examples to be expressed analytically. For context, we give a brief survey of related estimates on integral means and coefficients of univalent functions.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Mar 7, 2013

References