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. (Hint: Use Schwarz reection to extend f to all of C.)

Solution

Consider the function¯g(z)=f(z¯)¯.

The function g(z) is analytic in the unit disk and equals 1 on the lower half of the unit circle.

Now let h(z)=(f(z)1)(g(z)1).

We have that h(z)=0 both on the lower and upper half of the unit circle.

By the maximum modulus principle, h(z) is identically equal to 0.

Therefore, for every z either f(z)=1 or g(z)=1 (or both).

Since functions f(z) and g(z) are continuous, one of them must be equal to 1 in some neighborhood of 0, and thus be equal to 1 identically