Convexity of arbitrary sets in the plane in a direction (Q2716194)

The authors discuss criteria for directional convexity of arbitrary (not necessarily open or compact) sets in the plane \(\mathbf C\). Sets of this type appear e.g. in the study of solvability of differential equations of infinite order in the class of holomorphic functions. The authors define convexity of a subset \(Q\) of \(\mathbf C\) in the direction \(\beta\in\mathbf C\setminus\{O \}\) with respect to a point \(\alpha\in \mathbf C\setminus Q\) in terms of the set of values of a continuous branch of the multivalued function \(\arg z\). A subset \(Q\) of \(\mathbf C\) is said to be convex in the direction \(\beta= re^{i\theta}\), \(r> 0\), with respect to a point \(\alpha\in\mathbf C\setminus Q\), if any of the following two equivalent conditions is satisfied: I. There exists a continuous branch \(A\) of \(\arg (z-a)\) on \(Q\) such that for all \(k\in \mathbf Z\) the interval \([\theta+ \frac {\pi}2+2\pi k\), \(\theta+\frac {3\pi}2+2 \pi k]\not \subset A(Q)\). II. There exist a simply connected domain \(\Omega\) with \(Q\subset\Omega\subset \mathbf C\setminus\{\alpha\}\), and a continuous branch \(A\) of \(\arg (z-\alpha)\) on \(\Omega\) such that \(2A(Q)\subset (\theta-3\pi, \theta+3 \pi)\).NEWLINENEWLINEFor the entire collection see [Zbl 0959.00006].