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 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.

Keywords

• differential subordination,starlike function,lemniscate of Bernoulli,sine,Janowski function,exponential function,rational function,hypergeometric function

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