In this paper, we derive several fuzzy differential subordination and fuzzy differential superordination results for analytic functions ${\mathcal{M}}_{\xi ,\beta }^{s,\gamma }$, which involve the extended Mittag-Leffler function and the Pascal distribution series. We also investigate and introduce a class ${\mathcal{M}\mathcal{B}}_{\xi ,\beta }^{F,s,\gamma }(\rho )$ of analytic and univalent functions in the open unit disc $\mathcal{D}$ by employing the newly defined operator ${\mathcal{M}}_{\xi ,\beta }^{s,\gamma }$. We determine a specific relationship of inclusion for this class. Further, we establish prerequisites for a function role in serving as both the fuzzy dominant and fuzzy subordinant of the fuzzy differential subordination and superordination, respectively. Some novel results that are sandwich-type can be found here.

Nowadays, the subclasses of analytic functions in terms of fuzzy subsets are studied by various scholars and some of these concepts are extended using the $q-$theory of functions. In this inspiration, we introduce certain subclasses of analytic function by using the notion of fuzzy subsets along with the idea of $q-$calculus. We present the $q-$extensions of the fuzzy spiral-like functions of a complex order. We generalize this class using the $q-$analogues of the Ruscheweyh derivative and Srivastava-Attiya operators. Various interesting properties are examined for the newly defined subclasses. Also, some previously investigated results are deduced as the corollaries of our major results.

In this paper, we introduce and investigate two new subclasses of the function class $\mathrm{\Sigma }$ of bi-univalent functions of complex order defined in the open unit disk, which are associated with the Hohlov operator, satisfying subordinate conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions in these new subclasses. Several (known or new) consequences of the results are also pointed out.