Convex and alpha-prestarlike subordination chains

Abstract

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.