Show that if f is an analytic function on the unit disk that

Show that if f is an analytic function on the unit disk that is real-valued on the circle |z| = 1. then f is constant.

Solution

As the upper half plane with the real axis as boundary can be identified with the unit disk and the boundary, Schwarz reflection principle applies in this case.

So we can extend the function analytically over C to obtain a bounded analytic function on C.

Hence f is constant