TITLE: Find all entire function $f$ such that $\lim_{z\to \infty}\left|\frac{f(z)}{z}\right|=0$
QUESTION [6 upvotes]: If $f$ is an entire function such that $\lim_{z\to \infty}\left|\frac{f(z)}{z}\right|=0$ then find the function $f$.

Replacing $z$ by $\frac{1}{z}$, we get $$\lim_{z\to 0}|zf(1/z)|=0$$This shows that $f(1/z)$ has removable singularity at $z=0$ , so $f(z)$ has removable singularity at $z=\infty$. As $f$ is entire so , $f$ must be constant.
Is it correct?

REPLY [3 votes]: Yes correct. Here I present another argument that uses Liouville's theorem : Let 
$$
g(z):=\frac{f(z)-f(0)}{z}
$$
Clearly $g$ is also entire and bounded and thus constant! Since 
$$
\lim_{z \to \infty} g(z) = 0
$$
then $f(z)=f(0)$.