Geometric interpretation of radius of concavity and certain radii results for a class of meromorphic univalent functions

Abstract

Radii of convexity and starlikeness provide significant insights into the geometric properties for analytic functions $f$ defined in the open unit disk $\mathbb{D}$ of the complex plane. It is well-known that if $f(\mathbb{D})$ is a convex or a starlike domain, then $f(|z|<r)$ is also so for each $0<r<1$. But, for meromorphic univalent functions with nonzero pole, if $\overline{\mathbb{C}}\setminus f(\D)$ is a convex domain, it is not true that $\overline{\mathbb{C}}\setminus f(|z|<r)$ is a convex domain for each $0<r<1$. Considering this fact, in Bhowmik, B., Biswas, S.: Distortion, Radius of Concavity and Several Other Radii Results for Certain Classes of Functions, Comput. Methods Funct. Theory, 25 (2025), 393–418, we defined radius of concavity for meromorphic functions with nonzero pole by using the analytic characterization for concave univalent functions. In this article, at first, we provide a geometric interpretation for the definition of radius of concavity. Furthermore, we compute radii of concavity, convexity, and starlikeness for the class $\mathcal{V}_p(\lambda)$, where $\mathcal{V}_p(\lambda)$ consists of functions $f$ that are meromorphic in $\mathbb{D}$, having a simple pole at $z=p$, $p\in (0,1)$ and satisfying the differential inequality $\left|(z/f(z))^2f'(z)-1\right|<\lambda$, $z\in\D$, $\lambda \in (0,1]$.