Some extensions of the Kantorovich inequality and statistical applications

Kantorovich gave an upper bound to the product of two quadratic forms, (X'AX) (X'A-1X), where X is an n-vector of unit length and A is a positive definite matrix. Bloomfield, Watson and Knott found the bound for the product of determinants X'AX X'A-1X where X is n - k matrix such that X'X = Ik. In this paper we determine the bounds for the traces and determinants of matrices of the type X'AYY'A-1X, X'B2X(X'BCX)-1 X'C2X(X'BCX)-1 where X and Y are n - k matrices such that X'X = Y'Y = Ik and A, B, C are given matrices satisfying some conditions. The results are applied to the least squares theory of estimation.