On the preservation of direction-convexity and the Goodman-Saff conjecture

Abstract

Let K(Q be the set of univalent functions in the unit disk D which are convex in the direction eip . We determine the set of analytic functions g in D which preserve K(9) under the Hadamard product, i.e., g * f e K(p) whenever f e X(d. This result contains as a special case the proof of a conjecture of Goodman and Saff about l((rp) and solves partially a multiplier problem concerning convex univalent harmonic functions in D, posed by Clunie and Sheil-Small.