Limit of quotient of products $\lim_{n\to \infty}\left|\frac{\prod_{z^n=1}f(z)^n}{\prod_{z^n=-1}f(z)^n}\right|$

Limit of quotient of products $\lim_{n\to
\infty}\left|\frac{\prod_{z^n=1}f(z)^n}{\prod_{z^n=-1}f(z)^n}\right|$

I have a following problem: Let $f$ be a holomorphic function such that
$|f(z)|>1$ for $|z|=1$. So what can we say about the limit $$\lim_{n\to
\infty}\left|\frac{\prod_{z^n=1}f(z)^n}{\prod_{z^n=-1}f(z)^n}\right|?$$
Thanks in advance for any help!