On the radius of concavity for certain classes of functions

TL;DR

This paper investigates the radius of concavity for various classes of analytic functions, including those with positive real part derivatives, fixed coefficients, linearly invariant families, and meromorphic analogues, providing bounds and specific radii.

Contribution

It establishes new radii of concavity for multiple classes of functions, including classes with fixed coefficients and meromorphic counterparts, expanding understanding of geometric properties.

Findings

Derived radii of concavity for classes P' and with fixed second coefficients.

Established lower bounds for the radius of concavity for classes involving al functions.

Computed the radius of concavity for the meromorphic analogue of al classes.

Abstract

Let $\mathcal{A}$ denote the class of all analytic functions $f$ defined in the open unit disc $\mathbb{D}$ with the normalization $f(0)=0=f'(0)-1$ and let $P'$ be the class of functions $f\in\mathcal{A}$ such that ${\rm{Re}}\,f'(z)>0$, $z\in\mathbb{D}$. In this article, we obtain radii of concavity of $P'$ and for the class $P'$ with the fixed second coefficient. After that, we consider linearly invariant family of functions, along with the class of starlike functions of order $1/2$ and investigate their radii of concavity. Next, we obtain a lower bound of radius of concavity for the class of functions $\mathcal{U}_0(\lambda)=~\{f\in\mathcal{U}(\lambda) : f''(0)=0\}$, where $$ \mathcal{U}(\lambda)=\left\{f\in\mathcal{A} : \left|\left(\frac{z}{f(z)}\right)^2f'(z)-1\right|<\lambda,~z\in \mathbb{D}\right\},\quad \lambda \in (0,1]. $$ We also investigate the meromorphic analogue of the class $\mathcal{U}(\lambda)$ and compute its radius of concavity.