Volterra integration operators form one of the most actively studied classes of operators in current research. In this talk, we seek to generalize them by using “bona fide” fractional calculus. The strategy is first to introduce a class analytic paraproducts on the space of analytic functions in the unit disk. The main advantage in introducing this family of paraproducts comes from the fact that it combines the advantages of the Riemann-Liouville approach and the “coefficient multiplier” approach. Namely, one circumvents the use of multivalued functions but still retains a Leibniz formula and an integral representation formula. Many open problems will be introduced for this new class of operators.