A remark on level curves for domains convex in one direction^†^‡

Abstract

Let D be a plane domain which is convex in the v-direction, i.e.t the intersection of D with each vertical line is connected (or empty). It has been an open question whether level curves of a domain convex in the v-direction bound a domain with the same property. In this note we construct an example which settles the question in the negative.Closely related is a family ? of analytic functions g in the unit disk with the property that g(0) = 0 and Re{(1 - z2)g(z)/z} ⋚ 0. For univalent functions we show that membership in ? is essentially characterized by the geometric condition that Im g(ei0)⋚ 0 for a.e. ? ? (0, ?) and Im g(ei0)⋚ 0 for a.e. We conclude with a coefficient theorem