  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.
