ACTA issues

## New Hardy-type integral inequalities

Atanu Manna

Acta Sci. Math. (Szeged) 86:3-4(2020), 467-491
250/2019

 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.)