Radius constants for functions associated with a limacon domain
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Let $\mathcal{A}$ be the collection of analytic functions $f$ defined in $\mathbb{D}:=\left\{\xi\in\mathbb{C}:|\xi|<1\right\}$ such that $f(0)=f'(0)-1=0$. Using the concept of subordination ($\prec$), we define \begin{align*} \mathcal{S}^*_{\ell}:= \left\{f\in\mathcal{A}:\frac{\xi f'(\xi)}{f(\xi)}\prec\Phi_{\scriptscriptstyle{\ell}}(\xi)=1+\sqrt{2}\xi+\frac{\xi^2}{2},\;\xi\in\mathbb{D}\right\}, \end{align*} where the function $\Phi_{\scriptscriptstyle{\ell}}(\xi)$ maps $\mathbb{D}$ univalently onto the region $\Omega_{\ell}$ bounded by the limacon curve \begin{align*} \left(9u^2+9v^2-18u+5\right)^2-16\left(9u^2+9v^2-6u+1\right)=0. \end{align*} For $0<r<1$, let $\mathbb{D}_r:=\{\xi\in\mathbb{C}:|\xi|<r\}$ and $\mathcal{G}$ be some geometrically defined subfamily of $\mathcal{A}$. In this paper, we find the largest number $\rho\in(0,1)$ and some function $f_0\in\mathcal{G}$ such that for each $f\in\mathcal{G}$ \begin{align*} \mathcal{L}_f(\mathbb{D}_r)\subset\Omega_{\ell} ~ ~ \text{\ for every }~ 0<r\leq\rho, \end{align*} and \begin{align*} \mathcal{L}_{f_0}(\partial\mathbb{D}_\rho)\cap\partial\Omega_{\ell}\neq\emptyset, \end{align*} where the function $\mathcal{L}_f:\mathbb{D}\to\mathbb{C}$ is given by \begin{align*} \mathcal{L}_f(\xi):=\frac{\xi f'(\xi)}{f(\xi)}, \quad f\in\mathcal{A}. \end{align*} Moreover, certain graphical illustrations are provided in support of the results discussed in this paper.
Abstract
Let $\mathcal{A}$ be the collection of analytic functions $f$ defined in $\mathbb{D}:=\left\{\xi\in\mathbb{C}:|\xi|<1\right\}$ such that $f(0)=f'(0)-1=0$. Using the concept of subordination ($\prec$), we define \begin{align*} \mathcal{S}^*_{\ell}:= \left\{f\in\mathcal{A}:\frac{\xi f'(\xi)}{f(\xi)}\prec\Phi_{\scriptscriptstyle{\ell}}(\xi)=1+\sqrt{2}\xi+\frac{\xi^2}{2},\;\xi\in\mathbb{D}\right\}, \end{align*} where the function $\Phi_{\scriptscriptstyle{\ell}}(\xi)$ maps $\mathbb{D}$ univalently onto the region $\Omega_{\ell}$ bounded by the limacon curve \begin{align*} \left(9u^2+9v^2-18u+5\right)^2-16\left(9u^2+9v^2-6u+1\right)=0. \end{align*} For $0<r<1$, let $\mathbb{D}_r:=\{\xi\in\mathbb{C}:|\xi|<r\}$ and $\mathcal{G}$ be some geometrically defined subfamily of $\mathcal{A}$. In this paper, we find the largest number $\rho\in(0,1)$ and some function $f_0\in\mathcal{G}$ such that for each $f\in\mathcal{G}$ \begin{align*} \mathcal{L}_f(\mathbb{D}_r)\subset\Omega_{\ell} ~ ~ \text{\ for every }~ 0<r\leq\rho, \end{align*} and \begin{align*} \mathcal{L}_{f_0}(\partial\mathbb{D}_\rho)\cap\partial\Omega_{\ell}\neq\emptyset, \end{align*} where the function $\mathcal{L}_f:\mathbb{D}\to\mathbb{C}$ is given by \begin{align*} \mathcal{L}_f(\xi):=\frac{\xi f'(\xi)}{f(\xi)}, \quad f\in\mathcal{A}. \end{align*} Moreover, certain graphical illustrations are provided in support of the results discussed in this paper.
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