Kantorovich-type inequalities for operators via D-optimal design theory

The Kantorovich inequality is zTAzzTA−1z ⩽ (M + m)2/(4mM), where A is a positive definite symmetric operator in Rd, z is a unit vector and m and M are respectively the smallest and largest eigenvalues of A. This is generalised both for operators in Rd and in Hilbert space by noting a connection with D-optimal design theory in mathematical statistics. Each generalised bound is found as the maxima of the determinant of a suitable moment matrix.