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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?

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1 Answer 1

up vote 0 down vote accepted

On the unit circle, you have $(1-\bar\alpha_i z)=z(\bar z - \bar\alpha_i)$, therefore, the product is positive (and strictly so, given your conditions). So, you just need $c$ positive.

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Oh.. You missed multiplication factor $z$ on the right hand side, but I see your conclusion is right. Thanks for simple but great observation.. –  Detectives Dec 6 '12 at 17:58

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