Abstract. The proofs of generalized Hardy, Copson, Bennett, Leindler-type, and Levinson integral inequalities are revisited. It is contemplated to establish new proof of these classical inequalities using probability density function. New integral inequalities of Hardy-type involving the $r^{th}$ order \emph {Generalized Riemann--Liouville}, \emph {Generalized Weyl}, \emph {Erdélyi--Kober}, \emph {$(k, \nu )$-Riemann--Liouville}, and \emph {$(k, \nu )$-Weyl fractional integrals} are established through a probabilistic approach. The Kullback--Leibler inequality has been applied to compute the best possible constant factor associated with each of these inequalities.

DOI: 10.14232/actasm-019-750-7

AMS Subject Classification (1991): 26D15; 26D10, 26A33

Keyword(s): Hardy's inequality for integrals, probability density function, Riemann--Liouville integral, Weyl integral, scale distribution, Kullback--Leibler inequality

received 19.12.2019, revised 16.4.2020, accepted 18.4.2020. (Registered under 250/2019.)