Certain differential superordinations using a multiplier transformation and Ruscheweyh derivative

In the present paper we define a new operator, by means of convolution product between Ruscheweyh derivative and the multiplier transformation $I(m,\lambda,l)$. For functions $f$ belonging to the class $\mathcal A$ we define the differential operator $IR_{\lambda,l}^m\colon\mathcal A\to\mathcal A$, $IR_{\lambda,l}^m(z):=(I(m,\lambda,l)\ast R^m)f(z)$, where $\mathcal A_n=\{f\in\mathcal H(U)\colon f(z)=z+a_{n+1}z^{n+1}+\dots,\ z\in U\}$ is the class of normalized analytic functions, with $\mathcal A_1=\mathcal A$. We study some differential superordinations regarding the operator $IR_{\lambda,l}^m$.