On some inequalities of Opial-type (Q1340184)

Suppose that \(p\geq 1\), \(q> 0\), and \(r_k\geq 0\) \((k= 0, 1,\dots, n- 1)\) are numbers which satisfy \(r_0+ r_1+\cdots+ r_{n- 1}= 1\). Let \(f\in C^n[a, b]\), \(n\in \mathbb{N}\), and let \(w\) be a continuous weight function on \([a, b]\). \textit{W.-S. Cheung} [Matematika 37, No. 1, 136-142 (1990; Zbl 0706.26015)] has found upper estimates of the ratio \[ \int^b_a w(x) \Biggl( \prod^{n- 1}_{k= 0} |f^{(k)}(x)|^{r_k}\Biggr)^p |f^{(n)}(x)|^q dx\Biggl/ \int^b_a w(x)|f^{(n)}(x)|^{p+ q} dx\tag{1} \] under some additional assumptions on \(f\) and \(w\). The aim of this paper is to show that there are better estimates of (1) from above.