On Keogh’s Length Estimate for Bounded Starlike Functions

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.