Let \(1< p< \infty\), let \(f\) and \(v\) be measurable nonnegative functions on \([0, \infty)\) with \(v\) locally integrable and let \[ I(f)= \sup_{0\leq g\downarrow} \Biggl( \int^\infty_0 fg dx\Biggr) \Biggl(\int^\infty_0 \Biggl({1\over x} \int^x_0 g dy\Biggr)^p v(x) dx\Biggr)^{-1/p}. \] The main result is the sharp two-sided estimate of \(I(f)\).
It allows to reduce some inequalities for non-increasing functions to modified inequalities for arbitrary measurable functions. In particular, this approach allows to find for the case in which \(0< p\leq q< \infty\), \(q\geq 1\), necessary and sufficient conditions on nonnegative measurable functions \(w\) and \(v\) for the following inequality \[ \Biggl(\int^\infty_0 \Biggl({1\over x} \int^x_0 g dy\Biggr)^q w(x)dx\Biggr)^{1/q}\leq C\Biggl(\int^\infty_0 \Biggl({1\over x} \int^x_0 g dy\Biggr)^p v(x)dx\Biggr)^{1/p} \] to be valid for all non-increasing nonnegative functions \(g\) with \(C\) independent of \(g\).