Some Jessen-Beckenbach inequalities
One of the main results offered is the following. Let E be a nonempty set, \({\mathcal A}\) an algebra of its subsets and L a linear class of functions \(f: E\to {\mathbb{R}}\) containing the constant function and the characteristic functions of elements of \({\mathcal A}\). Furthermore, let B be a linear functional on L satisfying \(B(1)=1\) and \(B(f)\geq 0\) if \(f\geq 0\), I a closed real interval, \(\phi: I\to J\) convex, \(F: J^ 2\to {\mathbb{R}}\) nondecreasing in its first variable and \(F^ 2(x)=F(x,x).\) If \(E_ 1\in {\mathcal A}\) is such that B is positive on the characteristic function of \(E\setminus E_ 1\), then, for any \(g\in L\) with \(\phi(g)\in L\), \[ F^ 2(B[\phi(g)])\geq \inf_{x\in I}F^ 2(B[\phi(gC_ 1+xC_ 2)]), \] where \(C_ 1\) and \(C_ 2\) are the characteristic functions of \(E_ 1\) and \(E\setminus E_ 1\), respectively. \{In line 2 of p. 633, the third m should be M.\}
scientific article; zbMATH DE number 4063445