Journal of the
Korean Mathematical Society

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008



J. Korean Math. Soc. 2022; 59(2): 353-365

Online first article March 1, 2022      Printed March 1, 2022

Copyright © The Korean Mathematical Society.

Radius constants for functions associated with a limacon domain

Nak Eun Cho, Anbhu Swaminathan, Lateef Ahmad Wani

Pukyong National University; Indian Institute of Technology Roorkee; Indian Institute of Technology Roorkee


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

\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\},

where the function $\Phi_{\scriptscriptstyle{\ell}}(\xi)$ maps $\mathbb{D}$ univalently onto the region $\Omega_{\ell}$ bounded by the limacon curve


For $0

\mathcal{L}_f(\mathbb{D}_r)\subset\Omega_{\ell} ~ ~ \text{\ for every }~ 0



where the function $\mathcal{L}_f:\mathbb{D}\to\mathbb{C}$ is given by

\mathcal{L}_f(\xi):=\frac{\xi f'(\xi)}{f(\xi)}, \quad f\in\mathcal{A}.

Moreover, certain graphical illustrations are provided in support of the results discussed in this paper.

Keywords: Subordination, radii problems, lemniscates, limacon, cardioid, nephroid

MSC numbers: 30C45, 30C80

Supported by: The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No.~2019R1I1A3A01050861). The second and third-named authors were also supported by the Project No. CRG/2019/000200/MS of Science and Engineering Research Board, Department of Science and Technology, New Delhi, India.