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## C. Results from Analytic Function Theory

### C.9 Application of the Poisson-Jensen Formula to Certain Rational Functions

Consider the biproper rational function given by

 (C.9.1)

is a integer number, and and are polynomials of degrees and , respectively. Then, due to the biproperness of , we have that .

Further assume that

(i)
has no zeros outside the open unit disk,
(ii)
does not vanish on the unit circle, and
(iii)
vanishes outside the unit disk at .

Define

 (C.9.2)

where and are polynomials.

Then it follows that

(i)
has no zeros in the closed unit disk;
(ii)
does not vanish on the unit circle;
(iii)
vanishes in the open unit disk at , where for ;
(iv)
is analytic in the closed unit disk;
(v)
does not vanish on the unit circle;
(vi)
has zeros in the open unit disk, located at .

We then have the following result

Lemma C.3  Consider the function defined in (C.9.2) and a point such that ; then

 (C.9.3)

where is the Poisson kernel defined in (C.8.18).

##### Proof

This follows from a straightforward application of Lemma C.2.

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