Ukr. Mat. Zh. - 2019. - 71, № 6. - pp. 739-747
The main object of this paper is to improve some known estimates for the classical Kantorovich operators. We obtain a quantitative Voronovskaya-type result in terms of the second moduli of continuity, which improves some previous results. In order to explain the nonmultiplicativity of the Kantorovich operators, we present a Chebyshev – Gr¨uss inequality. Two Gr¨uss –Voronovskaya theorems for Kantorovich operators are also considered.
Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 723-740
We discuss the Grass inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave majorant of the modulus of continuity, we obtain a Grass inequality for the functional $L(f) = H(f; x)$, where $H: C[a,b] \rightarrow C[a,b]$ is a positive linear operator and $x \in [a,b]$ is fixed. We apply this inequality in the case of known operators, for example, the Bernstein, Hermite-Fejer operator the interpolation operator, convolution-type operators. Moreover, we derive Grass-type inequalities using Cauchy's mean value theorem, thus generalizing results of Cebysev and Ostrowski. A Grass 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 Grass' inequality. In an appendix, we prove a new result concerning the absolute first-order moments of the classical Hermite-Fejer operator.
Quantitative convergence theorems for a class of Bernstein–Durrmeyer operators preserving linear functions
Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 913–922
We supplement recent results on a class of Bernstein–Durrmeyer operators preserving linear functions. This is done by discussing two limiting cases and proving quantitative Voronovskaya-type assertions involving the first-order and second-order moduli of smoothness. The results generalize and improve earlier statements for Bernstein and genuine Bernstein–Durrmeyer operators.