Fujiwara’s inequality for synchronous functions and its consequences

The theory of inequalities is an essential tool in all fields of the theoretical and practical sciences, especially in statistics and biometrics. Fujiwara’s inequality provides relationships between expected values of products of random variables. Special cases of this inequality include important classical inequalities such as the Cauchy–Schwarz and Chebyshev inequalities.

The purpose of this article is to prove Fujiwara’s inequality in abstract probability spaces with more general assumptions for random variables than those associated with monotonicity as found in the literature. This newly introduced property of random variables will be called synchronicity. As a consequence, we obtain inequalities originated by Chebyshev, Hardy–Littlewood–Pólya and Jensen–Mercer under new conditions.

The results are illustrated by some examples constructed for specific probability measures and specific random variables. In biometrics, the obtained inequalities can be used wherever there is a need to compare the characteristics of experimental data based on means or moments, and the data can be modeled using functions that are synchronous or convex.