In this paper we consider the $\mathscr{L}^{n}:\mathcal{A} \rightarrow \mathcal{A}$, \\ $\mathscr{L}^{n}f\left(z\right)=\left(1-\lambda\right)\mathscr{D}^{n}f\left(z\right)+\lambda I^{n}f\left(z\right)$ linear operator, where $\mathscr{D}^{n}$ is the S\v{a}l\v{a}gean differential operator and $I^{n}$ is the S\v{a}l\v{a}gean integral operator. We give some results and applications for differential subordinations and superordinations for analytic functions and we will determine some properties on admissible functions defined with the new operator.