On some generalized Opial type inequalities (Q2708942)

In this paper some Opial type inequalities involving functions of one and several independent variables are obtained. In particular, the author proved that for \(f_i, g_i \in U\), \(i = 1, \dots, n\), and \(F, G \) and their first order partial derivatives \(F'_i, G'_i\), \(i= 1, \dots, n\), which are nonnegative continuous and nondecreasing on \(\mathbb{R}^n_+\) with \(F(0, \dots, 0) = G(0, \dots, 0) = 0\), the following inequality holds: NEWLINE\[NEWLINE\begin{aligned} \int_0^b&\Big[F\Big(|Tf_1 (t)|, \dots, |Tf_n (t)|\Big)\sum^n_{i=1}G'_i \Big(|Tg_1 (t)|, \dots, |Tg_n (t)|\Big)|g_i (t)|\\ &+ G\Big(|Tg_1 (t)|, \dots, |Tg_n (t)|\Big)\sum^n_{i=1}F'_i \Big(|Tf_1 (t)|, \dots, |Tf_n (t)|\Big)|f_i (t)|\Big]\\ &\leq F\Big(\int^b _a |f_1 (t)|dt,\dots, \int^b _a|f_n (t)|dt \Big) \cdot G \Big(\int^b _a |g _1 (t) |dt,\dots, \int ^b _a |g _n (t)|dt \Big) \end{aligned} \tag{1}NEWLINE\]NEWLINE where NEWLINE\[NEWLINETf(t) = \int^b_a f(s) ds \tag{2} NEWLINE\]NEWLINE is the Hardy operator and \(U\) is the class of functions which can be represented in the form (2).NEWLINENEWLINEFor the entire collection see [Zbl 0947.00027].