**Arnon Ploymukda and Pattrawut Chansangiam \***

Department of Mathematics, Faculty of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand; arnon.p.math@gmail.com

**\*** Correspondence: pattrawut.ch@kmitl.ac.th; Tel.: +66-935-266600

Received: 14 August 2019; Accepted: 5 October 2019; Published: 9 October 2019

**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.

**Keywords:** Chebyshev inequality; Tracy-Singh product; continuous field of operators; Bochner integral; weighted Pythagorean mean
