Applications of fuzzy differential subordination theory on analytic $p$ -valent functions connected with $\mathfrak{q}$-calculus operator

In recent years, the concept of fuzzy set has been incorporated into the field of geometric function theory, leading to the evolution of the classical concept of differential subordination into that of fuzzy differential subordination. In this study, certain generalized classes of $p$ -valent analytic functions are defined in the context of fuzzy subordination. It is highlighted that for particular functions used in the definitions of those classes, the classes of fuzzy $p$-valent convex and starlike functions are obtained, respectively. The new classes are introduced by using a $\mathfrak{q}$-calculus operator defined in this investigation using the concept of convolution. Some inclusion results are discussed concerning the newly introduced classes based on the means given by the fuzzy differential subordination theory. Furthermore, connections are shown between the important results of this investigation and earlier ones. The second part of the investigation concerns a new generalized $\mathfrak{q}$-calculus operator, defined here and having the $(p,\mathfrak{q}\mathfrak{)}$-Bernardi operator as particular case, applied to the functions belonging to the new classes introduced in this study. Connections between the classes are established through this operator.