Number of zero's of holomorphic function

[hedipaldi]

Homework Statement

Let f(z) be holomorphic in the open unit disc taking real values only on the real axis.show that f has at most one zero in the open disc.

Homework Equations

Rouche's theorem,the argument principle.

The Attempt at a Solution

obviously,f does not have non-real zero's .I need help in counting the zero's by using the argument principle.

 

[Dick]

Homework Statement

Let f(z) be holomorphic in the open unit disc taking real values only on the real axis.show that f has at most one zero in the open disc.

Homework Equations

Rouche's theorem,the argument principle.

The Attempt at a Solution

obviously,f does not have non-real zero's .I need help in counting the zero's by using the argument principle.

I'm not an expert, but is argument is zero at +1 and -1 and can't be zero anyplace else as the contour winds around the boundary of the unit circle. What are the possibilities for the winding number?