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Specific Classes of Analytic Functions Communicated with a Q-Differential Operator Including a Generalized Hypergeometic Function

A special function is a function that is typically entitled after an early scientist who studied its features and has a specific application in mathematical physics or another area of mathematics. There are a few significant examples, including the hypergeometric function and its unique species. These types of special functions are generalized by fractional calculus, fractal, q-calculus, (q,p)-calculus and k-calculus. By engaging the notion of q-fractional calculus (QFC), we investigate the geometric properties of the generalized Prabhakar fractional differential operator in the open unit disk ∇:={ξ∈C:|ξ|<1}. Consequently, we insert the generalized operator in a special class of analytic functions. Our methodology is indicated by the usage of differential subordination and superordination theory. Accordingly, numerous fractional differential inequalities are organized. Additionally, as an application, we study the solution of special kinds of q–fractional differential equation.