New results concerning Chebyshev–Grüss-type inequalities via discrete oscillations
Introduction
In the last years, Grüss-type inequalities have attracted much attention because of their applications [2], [4], [6], [7], [12], [13], [14], [16]. The classical form of the Grüss inequality gives an estimate of the difference between the integral of the product and the product of the integrals of two functions in . It was first published by G. Grüss in 1935 [10]. The functional given bywhere are integrable functions, is well known in the literature as the Chebyshev functional (see [5]). Theorem 1 Grüss, 1935, see [10] Let be integrable functions from into , such that , for all , where . Then
A special form of a theorem introduced by D. Andrica and C. Badea (see [3]), presenting an inequality for functionals, is given in the sequel. Theorem 2 Let be a compact interval of the real axis, the space of real-valued and bounded functions defined on I and L a linear positive functional satisfying where . Assuming that for one has for all , the following holdswhere .
Using the least concave majorant of the modulus of continuity, the authors of [1] obtained a Grüss-type inequality for the functional , where is a positive linear operator and is fixed. B. Gavrea and I. Gavrea [6] were the first to observe the possibility of using moduli in this context. In [17] a Grüss-type inequality in was proved, where is the set of continuous functions defined on a compact metric space . A functional is considered, where is a positive linear operator reproducing constant functions and is fixed. In order to formulate the result given in [17] we need the following. Definition 1 Let . If, for , the quantityis the usual modulus of continuity, then its least concave majorant is given byand is the diameter of the compact space X.
We callthe generalized Chebyshev functional and we will use the terminology Chebyshev–Grüss-type inequalities, referring to Grüss-type inequalities for generalized Chebyshev functionals. These inequalities have the general formwhere E is an expression in terms of certain properties of and some kind of oscillations of f and g.
In [17] the following Chebyshev–Grüss-type inequality which involves the least concave majorant of the modulus of continuity was proved. Theorem 3 see Theorem 3.1, in [17] If , where is a compact metric space, and , then the inequalityholds, where is the least concave majorant of the usual modulus of continuity and is the second moment of the operator H.
For , a slightly improved result was given in [17]: Theorem 4 see Theorem 4.1. in [17] If and is fixed, then the inequalityholds. Here we denote , for . Remark 1 It was shown in [9] that the inequality (2) is sharp in the sense that there exists a positive linear operator reproducing constant and linear functions, and there exist functions such that equality occurs.
In the following sections we will give some results in the bivariate discrete (positive) linear functional case and apply them to tensor products of (positive) linear operators. In the end we give Chebyshev–Grüss-type inequalities via discrete oscillations for more than two functions. The last section of the article [1] was an inspiration for these last results.
Section snippets
A bivariate Chebyshev–Grüss inequality
In [9] the authors obtained a new Chebyshev–Grüss-type inequality which involves oscillations of functions. This result is better than (2) in the sense that the oscillations of functions are relative only to certain points, while in (2) the oscillations, expressed in terms of , are relative to the whole interval . In this section we will give a generalization of the results obtained in [9], considering the bivariate discrete linear functional case.
Let X be an arbitrary set and the
Applications
We will apply the Chebyshev–Grüss-type inequality obtained in the previous section to known operators, for example Lagrange, Bernstein, Mirakjan–Favard–Szász and piecewise linear interpolation operators. Letwhere is fixed and H is a linear operator.
Chebyshev–Grüss-type inequalities via discrete oscillations for more than two functions
In [1], a Chebyshev–Grüss-type inequality on a compact metric space for more than two functions was introduced. We obtain a similar result, using the new approach implying discrete oscillations. This result is better than what was obtained in [1] in the sense that the oscillations of functions are relative only to certain points, while in [1] the oscillations are relative to the whole compact metric space X. Moreover, in what follows X is an arbitrary set, the set of all real-valued,
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