<p>The differential subordinations theory is also known as the admissible functions method and it was initiated by S.S. Miller and P.T. Mocanu in two publications in 1978 and 1981. The same two reputed researchers have introduced the notion of differential superordination as a dual notion of differential subordination in 2003. Generalizing the differential subordinations, J.A. Antonino and S. Romaguera introduced the notion of strong differential subordination in 1994. G.I. Oros has introduced the notion of strong differential superordination as a dual notion of strong differential subordination in 2009. Using the differential subordinations and superordinations, strong differential subordinations and superordinations methods it was easier to prove some classical results from this domain, and also many extensions or generalizations of these, at the same time new results being obtained. A useful tool in the study of different types of operators is offered by the theory of classical differential subordinations and superordinations and the theory of strong differential subordinations and superordinations. Some remarkable results in the differential and integral operators theory were successfully proved and new results were also obtained using those theories. In this article we establish several properties for the operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( IR_{\lambda ,l}^{m,n}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>I</mi> <msubsup> <mi>R</mi> <mrow> <mi>λ</mi> <mo>,</mo> <mi>l</mi> </mrow> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msubsup> </mrow> </math></EquationSource> </InlineEquation> defined as the Hadamard product of Ruscheweyh operator and multiplier transformation by using the classic and strong theories of differential subordinations and superordinations and get differential sandwich-type theorems. Interesting corollaries follow for particular functions used as best subordinant and best dominant.</p>