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We find sharp upper bounds for $ \vert \,f^{\prime \prime }(z)/f^{\prime }(z) \vert $ such that the function f be starlike or strongly-starlike of a given order, where f is holomorphic in the unit disc U , with $ f(0)=f^{\prime }(0)-1=0 $ . These bounds are obtained by using the differential subordination technique. As a useful ingredient we obtain the radius of convexity for the function $ F(z)= z /(\exp (z)-1) $ [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider convex subordination chains (c.s.c.) and alpha-prestarlike subordination chains (α-p.s.c.) over on the unit disc in the complex plane. We obtain sufficient conditions for a mapping to be an α-p.s.c. over , 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 on the unit disc. [Copyright &y& Elsevier]