On the radius of concavity for certain classes of functions

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 $\mathcal{U}_0(λ)=~\{f\in\mathcal{U}(λ) : f''(0)=0\}$, where $$ \mathcal{U}(λ)=\left\{f\in\mathcal{A} : \left|\left(\frac{z}{f(z)}\right)^2f'(z)-1\right|<λ,~z\in \mathbb{D}\right\},\quad λ\in (0,1]. $$ We also investigate the meromorphic analogue of the class $\mathcal{U}(λ)$ and compute its radius of concavity.