Angular derivatives and semigroups of holomorphic functions

Abstract

A simply connected domain $\Omega \subset \mathbb{C}$ is convex in the positive direction if for every $z\in \Omega$, the half-line $\{z+t:t\ge 0\}$ is contained in $\Omega$. We provide necessary and sufficient conditions for the existence of an angular derivative at $\infty$ 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.