Some Special Differential Subordinations

For an analytic function p satisfying $p(0)=1$, we obtain sharp estimates on β such that the first order differential subordination p(z)+β zp'(z) P(z) or 1+β zp'(z)/pʲ(z) P(z), $(j=0,1,2)$ implies p(z) Q(z) where P and Q are Carathéodory functions. The key tools in the proof of main results are the theory of differential subordination and some properties of hypergeometric functions. Further, these subordination results immediately give sufficient conditions for an analytic function f to be in various well-known subclasses of starlike functions.