The question is as follows.
the rational function defined on complex plane $ \displaystyle R(z) = c \cdot \prod_{i=1}^{n} \frac{(z- \alpha_i)(1-\bar \alpha_i z)}{(z-\beta_i)(1-\bar \beta_i z)} $
where $ c$: real, $\alpha_i , \beta_i $ : constants with $ |\alpha_i| < 1, | \beta_i| <1 $ $\alpha_i, \beta_i $ need not be distinct, assume it is listed according to their multiplicity.
is easily seen to be have real value on unit circle $|z|=1$.
when this function becomes positive on unit circle? Find necessary and sufficient condition respect to $c, \alpha_i, \beta_i$.
attempts to a solution :
I first worked with a intution that answer is $R(z) = L(z)^2$ for some Linear fractional transformation that maps unit circle to real line(which can be fully described in general form), so that every root of $R$ should have even multiplicity.
But, I was not able to prove it ...
Can anybody give some suggestion about this?