Definition. A Schur function is defined as a complex function analytic and of modulus not exceeding unity in .
Theorem. The function
Suppose is positive real. Then for , rere is PR. Consequently, is minimum phase which implies all roots of lie in the unit circle. Thus is analytic in . Also,
Conversely, suppose is Schur. Solving Eq.(5) for gives
If is constant, then is PR. If is not constant, then by the maximum principle, for . By Rouche's theorem applied on a circle of radius , , on which , the function has the same number of zeros as the function in . Hence, is minimum phase which implies is analytic for . Thus is PR.