An integral inequality for derivatives of equimeasurable rearrangements (Q1260827)

Let \(f\) be a Lebesgue measurable and a.e. finite function on the interval \(I=[0,1]\); set \(d(y)=m \{x;f(x)>y\}\), here \(m\) is the Lebesgue measure on the real line \(\mathbb{R}\). Then the rearrangement \(f^*\) of \(f\) is defined on \(I\) by \(f^*(x) = \inf \{y:d(y) \leq x\}\) if \(0 \leq x<1\) and \(f^*(1) = \text{ess} \inf f\) \((\inf \emptyset = + \infty)\). The rearrangement \(f^*\) is a nonincreasing right continuous function, equimeasurable with \(f\), i.e., \(f\) and \(f^*\) are identically distributed, as random variables on \(I\). (A good reference for studies of the operator \(f \to f^*\) is \textit{J. V. Ryff} [J. Math. Anal. Appl. 31, 449-458 (1970; Zbl 0214.137)].) Among several results, the author gets: 1. (Corollary 2.3). Let \(f\) be continuous on \(I=[0,1]\). Then, for any \(\delta>0\), \(V(\delta;f^*) \leq V(\delta;f)\) (for a function \(h\) on \(I\), \(V(\delta;h) = \sup \sum^ n_{i=1} | h(b_ i) - h(a_ i) |\), where the supremum is taken over all \(0 \leq a_ 1 < b_ 1 \leq \cdots \leq a_ n<b_ n \leq 1\) with \(\sum^ n_{i=1} (b_ i-a_ i) \leq \delta)\). This implies an older theorem of Ryff, contained in the above mentioned paper, namely that \(f^*\) is absolutely continuous, whenever \(f\) is so. 2. (Theorem 3.1). Let \(f\) be a.e. differentiable on \(I\) and \(F(y_ 1,y_ 2)\) a Borel measurable function on \(\mathbb{R} \times [0,+ \infty)\) such that \(y_ 2 \mapsto F(y_ 1,y_ 2)\) is nondecreasing for each fixed \(y_ 1\). Then \[ \int^ 1_ 0 F \biggl( f^* (x),\bigl | f^{*'} (x) \bigr | \biggr) dx \leq \int^ 1_ 0 F \biggl( f(x),\bigl | f'(x) \bigr | \biggr) dx. \] The proof of this theorem is based upon Ryff's method (\S2 in the above mentioned paper). This last inequality, in the particular case in which \(F(y_ 1,y_ 2)\) depends only on \(y_ 2\), has been proved (using the same method) by \textit{K. M. Chong} [Can. J. Math. 27, 330-336 (1975; Zbl 0319.28013)].