The theory of second order di erential subordination of S. S. Miller and P. T. Mocanu [Di erential Subordinations, Dekker, New York, 2000] was recently extended to functions with xed initial coecient by R. M. Ali, S. Nagpal and V. Ravichandran [Second-order di eren- tial subordination for analytic functions with xed initial coecient, 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 sucient conditions related to starlike- ness, convexity, close-to-convexity of normalized analytic functions are derived.