RADIUS CONSTANTS FOR FUNCTIONS ASSOCIATED WITH A LIMACON DOMAIN

• Published in: Journal of the Korean Mathematical Society Vol. 59; no. 2; pp. 353 - 365
• Main Authors: Cho, Nak Eun, Swaminathan, Anbhu, Wani, Lateef Ahmad
• Format: Journal Article
• Language: Korean
• Published: 2022
• Subjects: Subordination; nephroid; lemniscates; limacon; cardioid; radii problems
• ISSN: 0304-9914

Let be the collection of analytic functions f defined in := {ξ ∈ ℂ : |ξ| < 1} such that f(0) = f'(0) - 1 = 0. Using the concept of subordination (≺), we define $$S^*_{\ell}\;:=\;\{f{\in}A:\;\frac{{\xi}f^{\prime}({\xi})}{f({\xi})}{\prec}{\Phi}_{\ell}(\xi)=1+{\sqrt{2}{\xi}}+{\frac{{\xi}^2}{2}},\;{\xi}{\in}{\mathbb{D}}\}$$, where the function ℓ(ξ) maps univalently onto the region Ωℓ bounded by the limacon curve (9u2 + 9v2 - 18u + 5)2 - 16(9u2 + 9v2 - 6u + 1) = 0. For 0 < r < 1, let r := {ξ ∈ ℂ : |ξ| < r} and be some geometrically defined subfamily of . In this paper, we find the largest number ∈ (0, 1) and some function f0 ∈ such that for each f ∈ f ( r) ⊂ Ωℓ for every 0 < r ≤ , and $${\mathcal{L} _{f_0}}({\partial}{\mathbb{D}_{\rho})\;{\cap}\;{\partial}{\Omega}_{\ell}\;{\not=}\;{\emptyset}$$, where the function f : → ℂ is given by $${\mathcal{L}}_f({\xi})\;:=\;{\frac{{\xi}f^{\prime}(\xi)}{f(\xi)}},\;f{\in}{\mathcal{A}}$$. Moreover, certain graphical illustrations are provided in support of the results discussed in this paper.