Hardy’s, Carleman’s and Related Inequalities

Abstract

is a proof due to J. E. Littlewood], Kaluza and Szego [7], Knopp [8] and A. M. Ostrowski [12]. The corresponding integral theorem is obtained in Knopp [8]: 143 D. S. Mitrinović et al., Inequalities Involving Functions and their Integrals and Derivatives © Kluwer Academic Publishers 1991 HARDY'S, CARLEMAN'S AND RELATED INEQUALITIES 147If also K(x) is a decreasing function of x, and A(x) = l:anK(nx),For 1 < p ~ 2, we havexii) (Th. 271, Hardy and Littlewood [20]) If p > 1 and GU) is the geometric mean of f over (0, x), then (2.12)