TITLE: Characterize entire functions $f$ such that $|f(z)| \leq |\sin(z)|$
QUESTION [10 upvotes]: I want to determine all entire functions $f$ such that $|f(z)| \leq |\sin(z)|$. I searched around on MathSEx and I found the following question from which I tried to get inspired but I think it differs substantially from my question: Characterizing nonconstant entire functions with modulus 1 on the unit circle 
Here's what I tried. It is not complete I am missing some cases.
First if $|f(z)| = |\sin(z)|$ then let $h(z)= \frac{f(z)}{\sin(z)}$. Then $h$ is analytic on $\mathbb{C}\setminus \left\{n \pi,n\in\mathbb Z\right\}$, and we also have that $|h|=1$ on the unit disc $D$. So $\exists c$ such that $h(z)=c$ with $|c|=1$. $h$ is continuous at all $n \pi$, so $h(n\pi)=c.\sin(n\pi)$; so we have that $\exists c$ such that $|c|=1$ and $f(z)=c.\sin(z)$.
Now I need to take care of the case $|f(z)| < |\sin(z)|$ and I don't know what to do. Should I try to apply the Schwarz lemma to this case since we would get $|h|<1$?
Any help or solution is welcomed.

REPLY [10 votes]: If $|f(z)|\leq|\sin z|$ for all $z$, then the quotient $h(z)=\frac{f(z)}{\sin z}$ is analytic at the complement of the zero set of $\sin z$, as you noted, but it is also analytic at the zeroes (or, rather, has removable singularities there)
Indeed, your inequality implies that whenever $\sin z_0$ is zero, then $f(z_0)$ is zero also: this and a little work will show that $h$ is bounded in a neighborhood of $z_0$.
We thus conclude that $h$ is entire and bounded by $1$. You can probably take it from here.

REPLY [2 votes]: The function $ h(z) $ must obey $ |h(z)| \le 1 $ on all of $ \mathbb{C} $, which means that $ h(z) $ is a bounded function. By Liouville's theorem, any bounded entire function must be constant, hence $ f(z) = \alpha \sin(z) $ for all $ |\alpha| \le 1 $ characterizes the set of functions you want.