An integral inequality (Q2770410)

Let \((X,{\mathcal A},\mu)\) be a measure space, and let \(S:X\to {\mathcal A}\) be a function of type (C), i.e., \(S\) satisfies the following three conditions: C1) \(x\notin S(x)\) for every \(x\in X;\) C2) if \(y\in S(x),\) then \(S(y)\subset S(x);\) C3) \(\{ (x,y)\); \(y\in S(x)\} \) is \( \mu \times \mu \) measurable. Then, for every \(n\in {\mathbf N}^{{\mathbf \star } },\) and every \(l_{i}\in {\mathbf N}^{{\mathbf \star }}\) (for \(i=1,\dots ,n)\) with \(p=l_{1}+\dots +l_{n}\) one associates a \(\mu ^{p}\)-measurable set \( V_{l_{1},\dots ,l_{n}}(X)\) such that NEWLINE\[NEWLINE \int_{V_{l_{1},\dots ,l_{n}}(X)}f d\mu ^{p}\leq \frac{1}{|Q|}\int_{X^{p}} f d\mu ^{p} NEWLINE\]NEWLINE for all nonnegative \(\mu ^{p}\)-integrable functions \(f:X^{p}\to {\mathbb R}.\) Here \(|Q|\) is a positive integer, whose exact value is determined via an intricate combinatorial equivalence relation. This extends previous results due to \textit{A. M. Fink} [``Wendroff's inequalities'', Nonlinear Anal., Theory Methods Appl. 5, 873-874 (1981; Zbl 0471.26008)], and \textit{L. Horváth}, [``Gronwall-Bellman type integral inequalities in measure spaces'', J. Math. Anal. Appl. 202, 183-193 (1996; Zbl 0859.26008)]. A sample (for \( f=g\times \dots \times g)\) is as follows: NEWLINE\[NEWLINE \int_{X}\left( g(x)\left( \int_{S(x)}g d\mu \right) ^{p-1}\right) d\mu (x)\leq \frac{1}{p}\left( \int_{X}g d\mu \right) ^{p} .NEWLINE\]