Radius of convexity of certain classes of functions defined by convolution

Abstract

Let $\mathcal{S}$ be the class of analytic univalent functions defined in the open unit disc $\mathbb{D}$ of the complex plane with the normalizations $f(0)=0$ and $f'(0)=1$. For $A\in (1,2]$, let $Co(A)$ denote the class of concave univalent functions defined in $\mathbb{D}$ with the opening angle $πA$ at infinity. In this article, by applying certain convolution techniques, we investigate the radius of convexity for the class $Co(A)\ast\mathcal{S}t(1/2)$, where $\mathcal{S}t(1/2)\subsetneq \mathcal{S}$ denotes the class of starlike functions of order $1/2$. Furthermore, we establish that the radius of convexity of the class $\mathcal{S}\ast\mathcal{S}t(1/2)$ is at least $0.19191$ (approximately). Here, `$\ast$' denotes the convolution (or Hadamard product) of two classes of functions.