I hope this is not too long. Thanks in advance!
Edit: I edited it for a great deal, most of the information was unnecessary.
Let us define a function $h(z) = f(z) + g(z)$. We know that $f(z)$ has $K$ roots for $z$ within the complex unit circle and $g(z)$ also has $K$ roots for $z$ in the complex unit circle.
Rouche's theorem states that if $|f(z)| > |g(z)|, ~ |z| = 1$, $h(z)$ has the same number of roots in the complex unit disc that $f(z)$ has. In my case, this inequality does not always hold, it depends on other coefficients.
My question is, does it matter that $|f(z)|$ can be greater, equal, or smaller than $|g(z)|$? My gut feeling tells me that this should not change the number of roots $h(z)$ has, namely $K$.