  The theory of first-order differential subordination developed by Miller and
Mocanu 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. Connections with previously known
results are pointed out.
