On the Angular Derivative of Comb Domains

Abstract

The angular derivative problem is to provide geometric conditions on the boundary of a simply connected domain $$\Omega $$ near $$\zeta \in \partial \Omega $$ which are equivalent to the existence of a non-zero angular derivative at $$\zeta $$ for the conformal map of $$\Omega $$ onto the half plane. Rodin and Warschawski (Math Z 153:1–17, 1977), proposed a conjecture regarding the existence of an angular derivative at $$+\,\infty $$ for a certain class of comb domains. The purpose of this paper is to show how a theorem of Burdzy (Math Z 192:89–107, 1986) can be used to give an affirmative answer to the necessity part of the Rodin–Warschawski conjecture on comb domains.