Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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RADIUS CONSTANTS FOR FUNCTIONS ASSOCIATED WITH A LIMACON DOMAIN

  • Cho, Nak Eun (Department of Applied Mathematics Pukyong National University) ;
  • Swaminathan, Anbhu (Department of Mathematics Indian Institute of Technology Roorkee) ;
  • Wani, Lateef Ahmad (Department of Mathematics Indian Institute of Technology Roorkee)
  • Received : 2021.04.10
  • Accepted : 2021.10.14
  • Published : 2022.03.01
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Abstract

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 𝚽ℓ(ξ) maps 𝔻 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 𝔻r := {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 𝓛f (𝔻r) ⊂ Ωℓ for every 0 < r ≤ 𝜌, and $${\mathcal{L} _{f_0}}({\partial}{\mathbb{D}_{\rho})\;{\cap}\;{\partial}{\Omega}_{\ell}\;{\not=}\;{\emptyset}$$, where the function 𝓛f : 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.

Keywords

Acknowledgement

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. The authors would like to express their gratitude to the anonymous referees for their thoughtful comments and efforts towards improving the paper.

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