Let $f$ be an entire non-constant complex function and let $A$ and $B$ be given positive real constants. Is it possible that $|f(z)| \le A + B\log{| z|}$ for all complex $z$ such that $| z| \ge 1$ ?
I've been trying to solve using the fact that since $f$ is continuous in $\{z; |z|\le 1\}$ there exists $M=\sup\{ |f(z); |z|\le 1\}$, and then somehow use the Cauchy's integral formula for derivatives. Any ideas?
Thanks