Angular derivatives and semigroups of holomorphic functions

Abstract

A simply connected domain Ω⊂C is convex in the positive direction if for every z∈Ω, the half-line {z+t:t≥0} is contained in Ω. We provide necessary and sufficient conditions for the existence of an angular derivative at ∞ for domains convex in the positive direction which are contained either in a horizontal half-plane or in a horizontal strip. This class of domains arises naturally in the theory of semigroups of holomorphic functions, and the existence of an angular derivative has interesting consequences for the semigroup.