Some special differential subordinations

Authors : Nisha BOHRA, Sushil KUMAR, V. RAVİCHANDRAN

Pages : 1017-1034

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Publication Date : 2019-08-08

Article Type : Research Paper

Abstract :For an analytic function $p$ satisfying $pinsert ignore into journalissuearticles values(0);=1$, we obtain sharp estimates on $\beta$ such that  the first order differential subordination $pinsert ignore into journalissuearticles values(z);+\beta zp`insert ignore into journalissuearticles values(z);\prec \mathcal{P}insert ignore into journalissuearticles values(z);$ or  $1+\beta zp`insert ignore into journalissuearticles values(z);/p^{j}insert ignore into journalissuearticles values(z);\prec \mathcal{P}insert ignore into journalissuearticles values(z);$, $insert ignore into journalissuearticles values(j=0,1,2);$ implies $pinsert ignore into journalissuearticles values(z);\prec \mathcal{Q}insert ignore into journalissuearticles values(z);$ 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