Optimum Factorization of the Non-negative Matrix Functions

The proposition about optimum factorization of a non-negative matrix function $f(\lambda)$ is generalized for the case where the unknown function $A(z)$ of class $H_2$ satisfies the inequality
$$ A(e^{-i\lambda})A^*(e^{-i\lambda})\leqq 2\pi f(\lambda) $$
instead of the usual equality
$$ A(e^{-i\lambda})A^*(e^{-i\lambda})=2\pi f(\lambda). $$