Suppose $U$ is an open connected subset of the complex plane, and $f(z)$ is a non-constant holomorphic function on $U$. Suppose there exists $M\geq0$ such that
$$ \varlimsup_{n\to \infty} |f(z_n)|\le M$$
for every sequence $z_n\in U$ converging to a point in $\partial U$ or $|z_n|\to\infty$. Prove that the $|f(z)| \lt M$ for all $z\in U$.
this problem is related to Show holomorphic function bounded if limit of |f(z)| as z→z0∈∂U is bounded, but the answer seems not working, construct (by maximum modulus principle) a sequence $z_n$ with $|f(z_n)|> M$ is not true because $U$ may be Unbounded
How to get this? also, does this conclusion follow if the assumption $|z_n|\to\infty$ is dropped ?
Many thanks!