Summary: In the present paper we establish several differential superordinations regarding the operator \(DR^m_\lambda\) defined by using Ruscheweyh derivative \(R^mf(z)\) and the generalized Sălăgean operator \(D^m_\lambda f(z),DR^m_\lambda :\mathcal A_n\to\mathcal A_n,DR^m_\lambda f(z)=(D^m_\lambda*R^m)f(z),z\in U\), where \(m,n\in\mathbb N,\lambda\geq 0\) and \(f\in\mathcal A_n,\mathcal A_n=\{f\in\mathcal H(U):f(z)=z+\sum^\infty_{j=n+1}a_j z^j,z\in U\}\). A number of interesting consequences of some of these superordination results are discussed. Relevant connections of some of the new results obtained in this paper with those in earlier works are also provided.