Entire functions satisfying $|f(z)|\geq |\sin z|^{10} $ for all $z\in\mathbb{C}$ How do I find  all the  entire functions $f(z)$ such that $|f(z)|\geq |\sin z|^{10} $ for all $z\in\mathbb{C}$
Can an entire function have essential singularity? My intuition says that "no".
But I am not sure.
Can anyone help me ? 

If $f$ has no zeros, then $g(z)=\dfrac{(\sin z)^{10}}{f(z)}$ is entire and bounded and so is a constant.
Therefore, $f$ is a constant multiple of $(\sin z)^{10}$.
If $f$ has zeros, then they must be a subset of the zeros of $\sin z$, which are $n\pi$ for $n\in\mathbb Z$.
If $f$ has finitely many zeros, you can remove them from $\sin z$ and conclude that $g$ is constant as before.
If $f$ has infinitely many zeros, perhaps more can be said about $f$ by considering the product expansion:
$$
\begin{align}
\sin z = z \prod_{n = 1}^\infty \Bigl( 1- \frac{z^2}{n^2\pi^2} \Bigr).
\end{align}
$$
The Weierstrass factorization theorem may be useful but perhaps it's too advanced or is a sledgehammer for this problem.
