Riemann-Liouville fractional derivative operator; generalized Mittag-Leffler function; Hardy-type inequalities; Laplace transform method; Volterra differintegral equations; fractional differential equations; fractional kinetic equations; Lebesgue integrable functions; Fox-Wright hypergeometric functions

In this paper, we study a certain family of generalized Riemann-Liouville fractional derivative operators of order and type, which were introduced and investigated in several earlier works. In particular, we derive various compositional properties, which are associated with Mittag-Leffler functions and Hardy-type inequalities for the generalized fractional derivative operator. Furthermore, by using the Laplace transformation methods, we provide solutions of many different classes of fractional differential equations with constant and variable coefficients and some general Volterra-type differintegral equations in the space of Lebesgue integrable functions. Particular cases of these general solutions and a brief discussion about some recently investigated fractional kinetic equations are also given.