Characterisation of Entire functions

Question

Is it possible to characterise all entire functions with the property that $$\vert f'(z) \vert < \vert f(z) \vert \text{ for all $z \in \Bbb C$}?$$

Perhaps using Liouville's theorem?

Answer 1

$f$ has no zeroes, therefore $f(z) = e^{g(z)}$ for some entire function $g$. It follows that $$ \vert g'(z) \vert < 1 \text{ for all }z \, . $$ Now use Liouville's theorem to conclude that $g(z) = az + b$ with constants $a, b \in \Bbb C$, $\vert a \vert < 1$.