Grüss-type and Ostrowski-type inequalities in approximation theory

Abstract

We discuss the Grüss inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave majorant of the modulus of continuity, we obtain the Grüss inequality for the functional L.f / D H.f I x/; where H W C OEa; b ! C OEa; b is a positive linear operator and x 2 OEa; b is fixed. We apply this inequality in the case of known operators, e.g., the Bernstein operator, the HermiteFejér interpolation operator, and convolution-type operators. Moreover, we deduce Grüss-type inequalities using the Cauchy mean-value theorem, thus generalizing results of Chebyshev and Ostrowski. The Grüss inequality on a compact metric space for more than two functions is given, and an analogous Ostrowski-type inequality is obtained. The latter, in turn, leads to one further version of the Grüss inequality. In the appendix, we prove a new result concerning the absolute first-order moments of the classic Hermite-Fejér operator.

GBS operators of bivariate Bernstein‐Durrmeyer–type on a triangle

“…After publication, this inequality attracted a great deal of interest. Acu et al applied this inequality for the first time on the spaces of continuous functions defined on a metric space. Gal and Gonska obtained the Grüss‐Voronovskaja–type estimate for Bernstein polynomials and a class of one parameter family of operators of Bernstein‐Durrmeyer–type introduced by Paltanea .…”

Baskakov‐Durrmeyer type operators involving generalized Appell Polynomials

“…Grüss first established an inequality where he obtained an error estimate of the integral of product of two functions with the product of integrals of the two functions. Acu et al determined some applications of Grüss inequality for several operators by using least concave majorant. After that Gonska and Tachev studied the Grüss‐type inequality using second order modulus of smoothness.…”

Certain positive linear operators with better approximation properties

“…Motivated by the results obtained by Acu et al,() many researchers were interested to give Grüss‐type inequality for positive linear operators and to provide Grüss‐Voronovskaja–type theorem (see Mohiuddine et al and Acar et al). In the following, we get a Grüss‐Voronovskaja–type theorem for operators.…”