Convex and alpha-prestarlike subordination chains

In this paper we consider convex subordination chains (c.s.c.) and alpha-prestarlike subordination chains (α-p.s.c.) over (0, 1] on the unit disc in the complex plane. We obtain sufficient conditions for a mapping f (z, t) to be an α-p.s.c. over (0, 1], which generalize a well-known result of Ruscheweyh [St. Ruscheweyh, Convolutions in Geometric Function Theory, Les Presses de l’Université de Montreal, 1982].We also obtain sufficient conditions for injectivity for nonanalytic mappings on the unit disc, and give certain examples of α-c.s.c. over (0, 1] on the unit disc. © 2006 Elsevier Inc. All rights reserved