The results of this paper deal with a few applications of the general theory of the differential subordinations and superordinations, and to obtain our results we use the general techniques of the differential subordination and superordination theory, which could be considered one of the newest techniques used in some topics of Geometric Function Theory. In our investigation we find new differential subordination and superordination results of an operator for meromorphic multivalent functions defined by convolution product with the Hurwitz-Lerch Zeta function, that is ${\mathrm{K}}_{d,p}^s(n,m)$ with $d,n,m\in \mathbb{R}\setminus {\mathbb{Z}}_0^{-}$, $s\in \mathbb{R}$. The main results are followed by a few corollaries containing some special cases, examples, and applications.