Some harmonic 𝑛-slit mappings

Abstract

The class S H S_H consists of univalent, harmonic, and sense-preserving functions f f in the unit disk, Δ \Delta , such that f = h + g ¯ f=h+\overline {g} where h ( z ) = z + ∑ 2 ∞ a k z k h(z)=z+\sum _2^\infty a_kz^k , g ( z ) = ∑ 1 ∞ b k z k g(z)=\sum _1^\infty b_kz^k . S H O S_H^O will denote the subclass with b 1 = 0 b_1=0 . We present a collection of n n -slit mappings ( n ≥ 2 ) (n \geq 2) and prove that the 2 2 -slit mappings are in S H S_H while for n ≥ 3 n \geq 3 the mappings are in S H O 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.