Generalizations of two inequalities involving Hermitian forms (Q1063659)

Let \(\lambda_ 1\) and \(\lambda_ N\) be, respectively, the greatest and smallest eigenvalues of the Hermitian matrix \(H=(h_{ij})\). For x with \((x,x)=1\) and H positive definite the Kantorovich inequality states \[ (1)\quad (\lambda_ 1+\lambda_ N)^ 2/4\lambda_ 1\cdot \lambda_ N\geq (x,Hx)(x,H^{-1}x)\geq 1. \] For \(| y_ i| \leq 1\), \(i=1,...,N\) consider the Hermitian matrix \(M(y):=(\delta_{ij}+(1- \delta_{ij})\bar y_ iy_ j)\) and the Hadamard products M(y)*H and \(M(y)*H^{-1}\). It is shown that the inequality (1) remains true for M(y)*H and \(M(y)*H^{-1}\) instead of H and \(H^{-1}\); furthermore the estimate \(\rho\) (M(y)*H)\(\leq \rho (H)\) for the spectral radius \(\rho\) (M(y)*H) of M(y)*H is given.