Integral inequalities related to Hardy's inequality (Q1063129)

Hardy's classical inequality \[ \int^{\infty}_{0}(x^{-1}F(x))^ pdx\leq (p/(p-1))^ p\int^{\infty}_{0}(f(x))^ pdx, \] where \(f\geq 0\), \(F(x)=\int^{x}_{0}f(t)dt\) and \(1<p<\infty\), is generalized. The theorems are essentially the integral analogues of the series inequalities of \textit{J. Németh} [Acta Sci. Math. 32, 295-299 (1971; Zbl 0226.26020)], but they include many existing integral inequalities.