Some Special Differential Subordinations

Abstract

For an analytic function $p$ satisfying $p(0)=1$, we obtain sharp estimates on $\beta$ such that the first order differential subordination $p(z)+\beta zp'(z)\prec \mathcal{P}(z)$ or $1+\beta zp'(z)/p^{j}(z)\prec \mathcal{P}(z)$, $(j=0,1,2)$ implies $p(z)\prec \mathcal{Q}(z)$ where $\mathcal{P}$ and $\mathcal{Q}$ are Caratheodory 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.