Applications of differential subordination for functions with fixed second coefficient to geometric function theory

The theory of second order differential subordination of S.S. Miller and P.T. Mocanu [Differential Subordinations, Dekker, New York, 2000] was recently extended to functions with fixed initial coefficient by R.M. Ali, S. Nagpal and V. Ravichandran [Second-order differential subordination for analytic functions with fixed initial coefficient, Bull. Malays. Math. Sci. Soc. (2) 34 (2011), 611-629] and applied to obtain several generalization of classical results in geometric function theory. In this paper, further applications of this subordination theory is given. In particular, several sufficient conditions related to starlikeness, convexity, close-to-convexity of normalized analytic functions are derived. Keywords and Phrases: Analytic functions, Starlike functions, Convex functions, Subordination, Fixed second coefficient.
1. Introduction For univalent functions f(z) = z + [[SIGMA].sub.n=2.sup.[infinity]] [a.sub.n] [z.sup.n] defined on D: {z [member of] C: |z| < 1}, the famous Bieberbach theorem shows that |[a.sub.2]| [less [...]