The results from this paper are related to the geometric function theory. In order to obtain them, we use the technique based on the properties of the differential subordination and superordination one of the newest techniques used in this field, we obtain some differential subordination and superordination results for multivalent functions defined by differintegral operator with $j$-derivatives ${\mathrm{\Im }}_p(\nu ,\rho ;\ell )f(z)$ for $\ell >0,\nu ,\rho \in \mathbb{R},$such that $(\rho -j)\ge 0,\nu >-\ell p(p\in \mathbb{N})$ in the open unit disk $U$. Differential sandwich result is also obtained. Also, the results are followed by some special cases and counter examples.

The results of this work have a connection with the geometric function theory and they were obtained using methods based on subordination along with information on $\mathfrak{q}$-calculus operators. We defined the $\mathfrak{q}$-analogue of multiplier- Ruscheweyh operator of a certain family of linear operators $I_{\mathfrak{q},\mu }^s(\lambda ,\ell )\mathfrak{f}(\varsigma )(s\in {\mathbb{N}}_0=\mathbb{N}\cup \{0\},\mathbb{N}=\left\{1,2,3,..\right\};\ell ,\lambda ,\mu \ge 0,0<\mathfrak{q}<1)$. Our major goal was to build some analytic function subclasses using $I_{\mathfrak{q},\mu }^s(\lambda ,\ell )\mathfrak{f}(\varsigma )$ and to look into various inclusion relationships that have integral preservation features.