Two new classes of the generalized Kantorovich inequalities and their applications (Q1191691)

In the Kantorovich inequality \[ x'Axx'A^{-1}x\leq ((\lambda_ 1+\lambda_ n)^ 2/4\lambda_ 1\lambda_ n),\leqno (1) \] \(A\) is an \(n\times n\) positive definite matrix, \(x\) is an \(n\)-vector such that \(x'x=1\), \(\lambda_ 1\) and \(\lambda_ n\) represent the greatest and least eigenvalue, respectively. Substituting an \(n\times p\) matrix \(X\) of rank \(p\) for a vector \(x\) and using a proper measure, we obtain generalized Kantorovich inequalities (GKIs) which are all the extensions of (1), i.e. when \(X\) degenerates into a vector, these GKIs imply the inequality (1). In other words, the inequality (1) is a particular case of GKIs. Using different measures, we may distinguish these GKIs and divide them into three classes. Finally, some applications to multivariate linear models are given.