Applications of Fractional -Calculus to Certain Subclass of Analytic -Valent Functions with Negative Coefficients

By making use of the concept of fractional -calculus, we firstly define -extension of the generalization of the generalized Al-Oboudi differential operator. Then, we introduce new class of -analogue of -valently closed-to-convex function, and, consequently, new class by means of this new general differential operator. Our main purpose is to determine the general properties on such class and geometric properties for functions belonging to this class with negative coefficient. Further, the -extension of interesting properties, such as distortion inequalities, inclusion relations, extreme points, radii of generalized starlikeness, convexity and close-to-convexity, quasi-Hadamard properties, and invariant properties, is obtained. Finally, we briefly indicate the relevant connections of our presented results to the former results.