Length of ray-images under conformal maps

Abstract:

Let $w = f(z)$ be regular and univalent in $|z| < 1$ with $f(0) = 0$. Suppose that $f$ maps the unit disc onto a domain $D$. Let $l(r,\theta )$ be the length of the image curve of the ray joining $z = 0$ to $z = r{e^{i\theta }}$ in $D$ and $A(r) = \operatorname {Sup}[{\left | {f(r{e^{i\theta }})} \right |^{ - 1}}l(r,\theta )]$ where the supremum is taken over all starlike functions. In this paper we show that $A(r) \leqslant (1 + r)$.