An extension of Rouche's theorem

Pure mathematicians have paid insufficient attention to the properties of positive real functions which are of direct value to electrical network synthesis. It is for this reason that the author records an extension to Rouche's Theorem. Rouche proved in 1862 that if f(z) and g(z) are two functions regular within and on a closed contour C, on which f(z) does not vanish and in addition on C, g(z) < f(z) then f(z) and f(z) + g(z) have the same number of zeros within C. It is then possible to restate Rouche's theorem in form relating to the positive real character of the function F(z), which equals f(z) + g(z) divided by f(z). Since in addition F(z) is real for real z and is analytic in the right half plane, then it is a positive real function.