Computational Methods and Function Theory 5 (2005), No. 2, 263--274
Copyright Heldermann Verlag 2005
Edward Crane
edward.crane@gmail.com , Merton College, OX1 4JD, Oxford, U.K.
Dinesh Markose
dm295@dpmms.cam.ac.uk , Trinity College, CB2 1TQ Cambridge, U.K.
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.
Keywords: Starlike, conformal mappings, self similar, integral means.
MSC 2000: Primary 30C45; Secondary 30C35.
[FullText-pdf (265 K)] [FullText-ps (475 K)]