- Hacettepe Journal of Mathematics and Statistics
- Vol: 48 Issue: 4
- Some special differential subordinations
Some special differential subordinations
Authors : Nisha Bohra, Sushil Kumar, V. Ravichandran
Pages : 1017-1034
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Publication Date : 2019-08-08
Article Type : Research
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