I would like to verify my proof of the following statement:
Let $f(z)=z^n + O(z^{n-1})$ be a polynomial. Show that the Fatou set of $f$ does not contain a Herman ring.
Proof:
Suppose otherwise and let $U$ be a Herman ring. By the maximum modulus principle, $f$ attains a maximum on the boundary of $U$. Let $z_0\in \partial U$ be the maximum value. Since $f$ is entire, by maximum modulus principle again, $f$ is constant. This gives a contradiction.
Clearly there is something wrong with this proof; Any advice?
Your conclusion that $f$ is constant is incorrect (if you pick any non-constant polynomial $f$, and any bounded open set $U$, $|f|$ will indeed attain its maximum on $\partial U$).
Hint: you should apply the maximum principle not just to $f$, but to all of $f^n$, and not on the Herman ring, but on the bounded Jordan domain defined by a simple closed curved in the Herman ring.