Differential sandwich theorems involving Riemann-Liouville fractional integral of $q$-hypergeometric function

The development of certain aspects of geometric function theory after incorporating fractional calculus and $q$-calculus aspects is obvious and indisputable. The study presented in this paper follows this line of research. New results are obtained by applying means of differential subordination and superordination theories involving an operator previously defined as the Riemann-Liouville fractional integral of the $q$-hypergeometric function. Numerous theorems are stated and proved involving the fractional $q$-operator and differential subordinations for which the best dominants are found. Associated corollaries are given as applications of those results using particular functions as best dominants. Dual results regarding the fractional $q$-operator and differential superordinations are also considered and theorems are proved where the best subordinants are given. Using certain functions known for their remarkable geometric properties applied in the results as best subordinant, interesting corollaries emerge. As a conclusion of the investigations done by applying the means of the two dual theories considering the fractional $q$-operator, several sandwich-type theorems combine the subordination and superordiantion established results.