**1. Introduction**

One of the fundamental inequalities in mathematics is the Chebyshev inequality, named after P.L. Chebyshev, which states that

$$\frac{1}{n}\sum\_{i=1}^{n}a\_{i}b\_{i}\ \geqslant \left(\frac{1}{n}\sum\_{i=1}^{n}a\_{i}\right)\left(\frac{1}{n}\sum\_{i=1}^{n}b\_{i}\right)\tag{1}$$

for all real numbers *ai*, *bi* (1 *i n*) such that *a*1 ... *an* and *b*1 ... *bn*, or *a*1 - ... - *an* and *b*1 - ... - *bn*. This inequality can be generalized to

$$\sum\_{i=1}^{n} w\_i a\_i b\_i \quad \triangleright \quad \left(\sum\_{i=1}^{n} w\_i a\_i \right) \left(\sum\_{i=1}^{n} w\_i b\_i \right) \tag{2}$$

where *wi* - 0 for all 1 = 1, ... , *n*. A matrix version of (2) involving the Hadamard product was obtained in [1]. if

A continuous version of the Chebyshev inequality [2] says that *f* , *g* : [*a*, *b*] → R are monotone functions in the same sense and *p* : [*a*, *b*] → [0, ∞) is an integrable function, then

$$
\int\_{a}^{b} p(\mathbf{x})d\mathbf{x} \int\_{a}^{b} p(\mathbf{x})f(\mathbf{x})\mathbf{g}(\mathbf{x})d\mathbf{x} \gg \int\_{a}^{b} p(\mathbf{x})f(\mathbf{x})d\mathbf{x} \cdot \int\_{a}^{b} p(\mathbf{x})\mathbf{g}(\mathbf{x})d\mathbf{x}.\tag{3}
$$

*Symmetry* **2019**, *11*, 1256; doi:10.3390/sym11101256  www.mdpi.com/journal/symmetry

If *f* and *g* are monotone in the opposite sense, the reverse inequality holds. In [3], Moslehian and Bakherad extended this inequality to Hilbert space operators related with the Hadamard product by using the notion of synchronous Hadamard property. They also presented integral Chebyshev inequalities respecting operator means.

The Grüss inequality, first introduced by G. Grüss in 1935 [4], is a complement of the Chebyshev inequality. This inequality gives a bound of the difference between the product of the integrals and the integral of the product for two integrable functions. For each integral function *f* : [*a*, *b*] → R, let us denote

$$\mathcal{T}(f) = \frac{1}{b-a} \int\_{a}^{b} f(x) dx.$$

The Grüss inequality states that if *f* , *g* : [*a*, *b*] → R are integrable functions and there exist real constants *k*, *K*, *l*, *L* such that *k f*(*x*) *K* and *l g*(*x*) *L* for all *x* ∈ [*a*, *b*], then

$$|\mathcal{Z}(f\mathcal{g}) - \mathcal{Z}(f)\mathcal{Z}(\mathcal{g})| \ll \frac{1}{4}(K - k)(L - l). \tag{4}$$

This inequality has been studied and generalized by several authors; see [5–7]. In [7], the term Chebyshev-Grüss inequalities is used mentioning to Grüss inequalities for Chebyshev functions *T*I which defined as

$$T\_{\mathcal{T}}(f,\emptyset) := \mathcal{T}(f \cdot \emptyset) - \mathcal{T}(f) \cdot \mathcal{T}(\emptyset).$$

A general form of Chebyshev-Grüss inequalities is given by

$$|T\_{\mathcal{T}}(f,\mathcal{g})| \precleftarrow E(\mathcal{T}, f, \mathcal{g})$$

where *E* is an expression depending on the arithmetic integral mean I and oscillations of *f* and *g*. Chebyshev-Grüss inequalities for some kind of operator via discrete oscillations is presented by Gonska, Raça and Rusu [7].

On the other hand, the notion of tensor product of operators is a key concept in functional analysis and its applications particularly in quantum mechanics. The theory of tensor product of operators has been investigate in the literature; see, e.g., [8,9]. In [10,11], the authors extend the notion of tensor product to the Tracy-Singh product for operators on a Hilbert space, and supply algebraic/order/analytic properties of this product.

In this paper, we establish a number of integral inequalities of Chebyshev type for continuous fields of Hermitian operators relating Tracy-singh products and weighted Pythagorean means. The Pythagorean means considered here are three classical means -the geometric mean, the arithmetic mean, and the harmonic mean. The continuous field considered here is parametrized by a locally compact Hausdorff space Ω endowed with a finite Radon measure. In Section 2, we give basic results on Tracy-Singh products for Hilbert space operators and Bochner integrability of continuous field of operators on a locally compact Housdorff space. In Section 3, we provide Chebyshev type inequalities involving Tracy-Singh products of operators under the assumption of synchronous Tracy-Singh property. In Section 4, we establish Chebyshev integral inequalities concerning operator means and Tracy-Singh products under the assumption of synchronous monotone property. Finally, we prove Chebyshev-Grüss inequalities via oscillations for continuous fields of operators in Section 5. In the case that Ω is a finite space with the counting measure, such integral inequalities reduce to discrete inequalities. Our results include Chebyshev-type inequalities concerning tensor product of operators and Tracy-Singh/Kronecker products of matrices.
