rational function with special properties on unit disk.

Question

I'm now solving the following complex analysis problem.

"determine the form of rational function in a plane hat has a positive value on unit circle."

hint suggested me that such a rational function must have same number of zeros and poles inside unit disk. But I cannot understand that. Using symmetry(Schwarz reflection principle) I can see that number of poles(zeros) inside unit disk is equal to number of poles(zeros) outside unit disk. BUT I CANNOT SEE WHY NUMBER OF POLES AND ZEROS MUST BE EQUAL..

can anybody give me some suggestion?

Answer 1

Let $g(t) = \log(f(e^{i t}))$ for $t \in [0, 2 \pi]$. Then $g$ is well-defined, real valued, $g(0) = g(2 \pi)$ and

$$ g'(t) = \frac{i \, f'(e^{i t}) \, e^{i t}}{f(e^{i t})}. $$

Let $\gamma$ denote the unit circle then

$$ 0 = \frac{1}{2 \pi i}\int_0^{2 \pi} g'(t) dt = \frac{1}{2 \pi i} \int_{\gamma}\frac{f'(z)}{f(z)}dz $$

and the latter is equal to the number of zeroes minus the number of poles of $f$ on the unit disc.