Further, these subordination results immediately give sufficient conditions for an analytic function $f$ to be in various well-known subclasses of starlike functions.
📌 3 Key Takeaways
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Objective & Hypothesis: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.
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Key Findings & Data:The key tools in the proof of main results are the theory of differential subordination and some properties of hypergeometric functions.
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Conclusion & Impact:Further, these subordination results immediately give sufficient conditions for an analytic function $f$ to be in various well-known subclasses of starlike functions.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:differential subordination · starlike function · lemniscate of Bernoulli
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.
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