RADIUS CONSTANTS FOR FUNCTIONS ASSOCIATED WITH A LIMACON DOMAIN

Let 𝓐 be the collection of analytic functions f defined in 𝔻 := {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 𝚽_ℓ(ξ) maps 𝔻 univalently onto the region Ω_ℓ bounded by the limacon curve (9u² + 9v² - 18u + 5)² - 16(9u² + 9v² - 6u + 1) = 0. For 0 < r < 1, let 𝔻_r := {ξ ∈ ℂ : |ξ| < r} and 𝒢 be some geometrically defined subfamily of 𝓐. In this paper, we find the largest number 𝜌 ∈ (0, 1) and some function f₀ ∈ 𝒢 such that for each f ∈ 𝒢 𝓛_f (𝔻_r) ⊂ Ω_ℓ for every 0 < r ≤ 𝜌, and $${\mathcal{L} _{f_0}}({\partial}{\mathbb{D}_{\rho})\;{\cap}\;{\partial}{\Omega}_{\ell}\;{\not=}\;{\emptyset}$$, where the function 𝓛_f : 𝔻 → 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.