Gelfond-Schneider Theorem/Lemma 2

Lemma
Let $f \left({z}\right)$ be an analytic function in the disk $D \subseteq \C: D = \left\{{z : \left|{z}\right| < R}\right\}$ for some $R \in \R$.

Let $f$ also be continuous on the closure of $D$, that is, on $D^- = \left\{{z : \left|{z}\right| \le R}\right\}$.

Then:
 * $\forall z \in D^-: \left|{f \left({z}\right)}\right| \le \left|{f}\right|_R$