Convolution Properties of a Class of Analytic Functions

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 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 $\alpha\geq0,\,m\geq2$ $p_{_{i}}\prec h,\, i=1,2.$ Several convolution properties are obtained by using method differential subordination. Many relevant connections findings here those earlier works pointed out as special cases.