could you give me a hint on how to approach problems like this:
Prove or disprove the existence of non-zero entire function such that
$$ |f(z)|^2 \leq|\cos z|$$ for all $z$ in $C$.
I think this one is similar too:
Let $f$ is a meromorphic function and $|f(z)|^3 \leq |\tan z|$ for all $z$ in $C \setminus\{P(f)\}$, where P(f) is a set of poles of $f$ in $C$. Prove that $f \equiv0$. The only thing I see doing is assuming that there is not such a function and then trying to apply Liouville theorem for an entire function $g(z)=1/(f(x)$, but it doesn't help. Thank you.
Let $g(z) = f(z)^3 \cos z$ in your second question. $g$ is meromorphic and less than $\sin z$ in magnitude where it's defined.