Typically real mean univalent functions of large growth

Abstract

Typically real normalized κ-dimensionally mean univalent functionsf on |z|<1 are considered for which

$$\mathop {\lim \sup }\limits_{r \uparrow 1} \{ (1 - r)^2 \mathop {\max }\limits_{0 \leqslant \theta \leqslant 2\pi } \left| {f(re^{i\theta } )} \right|\} > 0$$

. Lets=logf(z) forz in the unit disc cut along (−1, 0]. A theorem is proved concerning the area of the Riemann surface over thes-plane which distinguishes the two cases −1≦κ<+∞ and κ=+∞.