Subordinations and superordinations studies using $q$-difference operator

The results of this work belong to the field of geometric function theory, being based on differential subordination methods. Using the idea of the $\mathfrak{q}$-calculus operators, we define the $\mathfrak{q}$-analogue of the multiplier- Ruscheweyh operator of a specific family of linear operators, $I_{\mathfrak{q},\mu }^s(\lambda ,\ell ).$ Our major goal is to build and investigate some analytic function subclasses using $I_{\mathfrak{q},\mu }^s(\lambda ,\ell )$. Also, some differential subordination and superordination results are obtained. Moreover, based on the new theoretical results, several examples are constructed. For every differential superordination under investigation, the best subordinant is provided.