Radius of concavity for certain class of functions

TL;DR

This paper determines the radius of concavity for meromorphic univalent functions with a pole at a fixed point, and explores linear combinations of such functions to analyze their geometric properties.

Contribution

It introduces the radius of concavity for the class al S(p) and investigates the univalence, convexity, and concavity radii of linear combinations of these functions.

Findings

Calculated the radius of concavity for al S(p).

Analyzed univalence, convexity, and concavity radii of linear combinations.

Extended results to other well-known classes of univalent functions.

Abstract

Let $ \mathcal{S}(p) $ be the class of all meromorphic univalent functions defined in the unit disc $ \mathbb{D} $ of the complex plane with a simple pole at $ z=p $ and normalized by the conditions $ f(0)=0 $ and $ f^{\prime}(0)=1 $. In this paper, we find radius of concavity and compute the same for functions in $ \mathcal{S}(p) $ and for some other well-known classes of functions on unit disk. We explore general linear combinations $F(z):=\lambda_1f_1(z)+\cdots+\lambda_{2n} f_{2n}(z),\; \lambda_j\in\mathbb{C} $, $ n\in\mathbb{N} $, of functions belonging to the class $\mathcal{S}(p)$ and some other classes of functions of analytic univalent functions and investigate their radii of univalence, convexity and concavity.