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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. 