# Classical Kantorovich operators revisited

## TL;DR

This paper refines estimates for classical Kantorovich operators, providing improved approximation results and new inequalities, including Voronovskaya-type theorems and a Chebyshev-Grüss inequality, to better understand their properties.

## Contribution

It introduces improved approximation estimates and new inequalities for classical Kantorovich operators, enhancing understanding of their behavior and non-multiplicativity.

## Key findings

- A new quantitative Voronovskaya-type result with second moduli of continuity.
- A Chebyshev-Grüss inequality explaining non-multiplicativity.
- Two Grüss-Voronovskaya theorems for Kantorovich operators.

## Abstract

The main object of this paper is to improve some of the known estimates for classical Kantorovich operators. A quantitative Voronovskaya-type result in terms of second moduli of continuity which improves some previous results is obtained. In order to explain non-multiplicativity of the Kantorovich operators a Chebyshev-Gr\"uss inequality is given. Two Gr\"uss-Voronovskaya theorems for Kantorovich operators are considered as well.