On the Angular Derivative of Comb Domains
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.
KeywordsConformal map Angular derivative Lipschitz graphs Extremal distance
Mathematics Subject Classification30C35 31A15
I would like to thank Prof. Don Marshall and Prof. Dimitrios Betsakos for all their help with this work. I would also like to thank the referees for their helpful suggestions.
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