This paper investigated the $\mathfrak{q}$-analogue of the multiplier-Ruscheweyh operator acting on meromorphic analytic functions, denoted by ${\mathfrak{D}}_{\mathfrak{q},\epsilon }^r(\varkappa ,\varrho ).$ By applying tools from $\mathfrak{q}$-calculus together with the principle of subordination, we developed several analytical results that deepened the understanding of geometric function theory (GFT) in the setting of meromorphic functions. The study focused on constructing new subclasses of meromorphic univalent functions associated with the operator ${\mathfrak{D}}_{\mathfrak{q},\epsilon }^r(\varkappa ,\varrho ),$ characterized by $\mathfrak{q}$-starlikeness, $\mathfrak{q}$-convexity, and related geometric classes. Various inclusion relationships, differential inequalities, and integral preservation properties were examined to establish the structural behavior of these families of functions. The findings generalized and unified several existing results in the literature concerning different operators and extended their applications to broader contexts within meromorphic function theory with $\mathfrak{q}$-calculus operator.

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.

This paper deals with the study of some classes of analytic functions in the open unit disk defined by using subordinations and connected with the distribution series, and these new classes reduces to some inequalities involving the first and second-order derivatives of the functions. For a particular choice of parameters, these new classes reduce to many others studied by different authors, and give an extension of many of these previously investigated classes. First, we determine several simple sufficient conditions for generalized distribution series that belong to the new defined classes $S(D,E;\delta ,\rho )$ and $K(D,E;\delta ,\rho )$, and we establish certain inclusion properties between the subclasses $\mathcal{H}(\lambda ,\eta )$ and $K(D,E;\delta ,\rho )$. In addition, we obtain sufficient conditions to show that some related series belong to certain subclasses of analytic functions. Each of the results are followed by some special cases that were recently obtained in different articles. Additionally, specific instances of our primary outcomes are briefly mentioned. The main tools for obtaining our new results consist of a few simple inequalities presented in the second section, while the scope of the paper is to extend and continue the studies connecting specific problems of analytic functions with generalized distribution series.