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.)
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