Integral Inequalities of Chebyshev Type for Continuous Fields of Hermitian Operators Involving Tracy�Singh Products and Weighted Pythagorean Means

Abstract

In this paper, we establish several integral inequalities of Chebyshev type for bounded continuous fields of Hermitian operators concerning Tracy-Singh products and weighted Pythagorean means. The weighted Pythagorean means considered here are parametrization versions of three symmetric means: the arithmetic mean, the geometric mean, and the harmonic mean. Every continuous field considered here is parametrized by a locally compact Hausdorff space equipped with a finite Radon measure. Tracy-Singh product versions of the Chebyshev-Grüss inequality via oscillations are also obtained. Such integral inequalities reduce to discrete inequalities when the space is a finite space equipped with the counting measure. Moreover, our results include Chebyshev-type inequalities for tensor product of operators and Tracy-Singh/Kronecker products of matrices.