Radius of concavity for certain class of 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 $F(z):={\lambda }_1{f}_1(z)+\cdots +{\lambda }_{2n}{f}_{2n}(z),{\lambda }_j\in \mathbb{C}$, $n\in \mathbb{N}$, 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.