The paper contains several results of the following type: Theorem. Let h be analytic in , let be analytic in a domain D containing h(U) and suppose a) Re (h(z)), and either b) h(z) is convex or b) is starlike w.r.t. the origin. If p is analytic in U with , p(U), and
then p(z). An application to univalent functions is given: Let f be analytic in U, and either
Then f is starlike in U.