On the roots of determinants of a class of holomorphic matrix-valued functions

We consider matrix-valued functions that are holomorphic in the unit disk that are Cauchy transforms of finite matrix-valued measures. For non-zero roots z j of the determinants of such functions, we provide estimates for ∑ ( | z j | − 1 − 1 ) and [ ∑ ( | z j | − 1 − 1 ) 2 ] 1 / 2 .