将单位圆盘映射到伯努利矩阵内部的解析函数

IF 1.3 Q3 COMPUTER SCIENCE, THEORY & METHODS
Shalu Yadav, Vaithiyanathan Ravichandran
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引用次数: 1

摘要

由\begin{document}$ \varphi_L(z) = \sqrt{1+z} $\end{document}定义的函数\begin{document}$ \varphi_L $\end{document}将单位磁盘\begin{document}$ \mathbb{D} $\end{document}映射到\begin{document}$ \Omega = \{w\in\mathbb{C}: |w^2-1|伯努利\begin{document}$ |w^2-1|右半平面上以伯努利\begin{document}$ |w^2-1|为界的区域$\end{document}。本文讨论了在\begin{document}$ \mathbb{D} $\end{document}上定义的星形函数与\begin{document}$ zf'(z)/f(z)\in \Omega $\end{document}或等价于\begin{document}$ zf'(z)/f(z) $ $ end{document}从属于\begin{document}$ \varphi_L(z) $\end{document},这些函数与解析函数\begin{document}$ p:\mathbb{D}\到\mathbb{C} $\end{document}与\begin{document}$ p(z)\in \mathbb{D}的所有\begin{document}$ z\in \mathbb{D}有关由\ \{文档}结束美元开始{文档}$ p (z) = zf ' (z) / f (z) $ \{文档}结束。利用一阶和二阶微分隶属性的可容许准则,研究了函数\begin{document}$ p $\end{document}满足\begin{document}$ p(z)\in \Omega $\end{document}的几个隶属性结果。作为应用,我们给出了函数\begin{document}$ f $\end{document}满足\begin{document}$ zf'(z)/f(z)\in \Omega $\end{document}的几个充分条件。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Analytic function that map the unit disk into the inside of the lemniscate of Bernoulli

The function \begin{document}$ \varphi_L $\end{document} defined by \begin{document}$ \varphi_L(z) = \sqrt{1+z} $\end{document} maps the unit disk \begin{document}$ \mathbb{D} $\end{document} onto \begin{document}$ \Omega = \{w\in\mathbb{C}: |w^2-1|<1\} $\end{document}, the region in the right half-plane bounded by the lemniscate of Bernoulli \begin{document}$ |w^2-1| = 1 $\end{document}. This paper deals with starlike functions defined on \begin{document}$ \mathbb{D} $\end{document} with \begin{document}$ zf'(z)/f(z)\in \Omega $\end{document} or equivalently \begin{document}$ zf'(z)/f(z) $\end{document} is subordinated to \begin{document}$ \varphi_L(z) $\end{document} and these functions are related to the analytic function \begin{document}$ p:\mathbb{D}\to \mathbb{C} $\end{document} with \begin{document}$ p(z)\in \Omega $\end{document} for all \begin{document}$ z\in \mathbb{D} $\end{document} by \begin{document}$ p(z) = zf'(z)/f(z) $\end{document}. Using the admissibility criteria of the first and second order differential subordination, we investigate several subordination results for functions \begin{document}$ p $\end{document} to satisfy \begin{document}$ p(z)\in \Omega $\end{document}. As applications, we give several sufficient conditions for functions \begin{document}$ f $\end{document} to satisfy \begin{document}$ zf'(z)/f(z)\in \Omega $\end{document}.

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