We focus on the improvement of operator Kantorovich type inequalities. Among the consequences, we improve the main result of the paper [H.R. Moradi, I.H. Gümüş, Z. Heydarbeygi, A glimpse at the operator Kantorovich inequality, Linear Multilinear Algebra, doi:10.1080/03081087.2018.1441799].
A set to locate all eigenvalues for matrices with a constant main diagonal entry is given, and it is proved that this set is tighter than the well-known Geršgorin set, the Brauer set and the set proposed in (Linear and Multilinear Algebra, 60:189-199, 2012). Furthermore, by applying this result to Toeplitz matrices as a subclass of matrices with a constant main diagonal, we obtain a set including all eigenvalues of Toeplitz matrices.