If f A01 C is a nonconstant continuous function that is ana

If f: A(0,1) - (C is a non-constant continuous function that is analytic in A(0,1) and satisfies |f(z)|f = 1 for every z on the circle k\"(0,1), demonstrate that/has the form for z in A(0,1), where a1,a2,...,ar are distinct points of the disk A(0,1), mi, m-2,..., mr are positive integers, and c is a constant of unit modulus.

Solution

Let . a[k]s be the distinct zeros of f(z) counted with multiplicity m[k]. Both have to be finite , as the function f is analytic inside the unit disc and cannot attain 0 on the unit disc.(as |f(z)| =1on the circle).

Let g(z) be the right hand side product.

Then h(z) =f(z)/g(z) is analytic inside the unit disc and |h(z) | =1 on the unit circle.

The same holds for 1/h(z).

Thus , by maximum modulus principle , |h(z)| =1 , thus f(z) =cg(z) with |c|=1, as required.