On Keogh's length estimate for bounded starlike functions

ET Crane, D Markose

Research output: Contribution to journalArticle (Academic Journal)peer-review


For a bounded starlike function f 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.
Translated title of the contributionOn Keogh's length estimate for bounded starlike functions
Original languageEnglish
Pages (from-to)263 - 274
Number of pages12
JournalComputational Methods and Function Theory
Volume5 (2)
Publication statusPublished - 2005

Bibliographical note

Publisher: Heldermann Verlag


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