Some harmonic $n$-slit mappings

Abstract:

The class $S_H$ consists of univalent, harmonic, and sense-preserving functions $f$ in the unit disk, $\Delta$, such that $f=h+\overline {g}$ where $h(z)=z+\sum _2^\infty a_kz^k$, $g(z)=\sum _1^\infty b_kz^k$. $S_H^O$ will denote the subclass with $b_1=0$. We present a collection of $n$-slit mappings $(n \geq 2)$ and prove that the $2$-slit mappings are in $S_H$ while for $n \geq 3$ the mappings are in $S_H^O$. Finally we show that these mappings establish the sharpness of a previous theorem by Clunie and Sheil-Small while disproving a conjecture about the inner mapping radius.