Chebyshev's inequality for functions whose averages are monotone (Q2367737)

Let an averaging operator \(A\) be defined by \(Af(x)=\int^ x_ 0 pf/\int^ x_ 0 p\), for suitable \(f\) and a continuous \(p>0\). Various generalizations of the well-known Chebyshev's inequality are obtained. For example, Theorem 1 gives the following result: If \((f-Af)(g-Ag)\geq 0\) for \(0\leq x\leq a\), Chebyshev's inequality \[ \int^ x_ 0pf\int^ x_ 0pg\leq \int^ x_ 0 p\int^ x_ 0 pfg \] holds for \(0\leq x\leq a\). Note that this result is equivalent to a result from a paper by \textit{M. Biernacki} [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 5, 23-29 (1952; Zbl 0048.289)].