On the product of convex combinations and the Kantorovich inequality
The paper deals with the problem to find the maximum and minimum of the product of two positive definite quadratic forms as a function on the unit sphere in n-dimensional complex space. If the two quadratic forms have a simultaneous diagonalization then the above problem is equivalent with the problem to maximize resp. minimize the product of two convex combinations of two sets of n real numbers with the same weights as a function of the weights. In this case a formula for the maximum and minimum is given thus relating to a classical inequality of Kantorovich for products of two convex combinations of positive numbers and their inverses with the same weights. In the general case estimates for the quotient between the maximal and minimal value are given.
scientific article; zbMATH DE number 4017104