Convolution Properties of a Class of Analytic Functions

Abstract

In this paper, we introduce a new class $\mathcal{R}^{\alpha}_{m}(h)$ of functions $F=f\ast\psi$, defined in the open unit disc $E$ with $F(0)=F^{\prime}(0)-1=0$ and satisfying the condition \begin{equation*} F^{\prime}(z)+\alpha zF^{\prime\prime}(z)=\left(\frac{m}{4}+\frac{1}{2}\right)p_{_{1}}(z)-\left(\frac{m}{4}-\frac{1}{2}\right)p_{_{2}}(z), \end{equation*} for $\alpha\geq0,\,m\geq2$ and $p_{_{i}}\prec h,\, i=1,2.$ Several convolution properties of this class are obtained by using the method of differential subordination. Many relevant connections of the findings here with those in earlier works are pointed out as special cases.