Differential inequalities for spirallike and strongly starlike functions

In this paper, by using a technique of the first-order differential subordination, we find several sufficient conditions for an analytic function p such that p(0) = 1 to satisfy Re{e(i beta)p(z)} > gamma or vertical bar arg{p(z) - gamma}vertical bar < delta for all z is an element of D, where beta is an element of(-pi/2, pi/2), gamma is an element of [0, cos beta), delta is an element of (0, 1] and D := {z is an element of C : vertical bar z vertical bar < 1}. The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in D.