I have to show by using Liouville's theorem, whether there are non-constant entire functions such that:
$$ |f^k(z)| \leq 1 \forall z \in \mathbb{C}$$ and fixed $k$.
- for $k=0$: There is such a function.
- for $k\geq1$: There shouldn't be such a function, because $f(z)$ is not necessarily bounded, isn't it?
Best Answer
If $f^{k}$ stands for the $k-$ th power then $|f^{k}(z)| \leq 1$ is same as $|f(z)| \leq 1$ provided $k \neq 0$. Hence $f$ is a constant if $k \neq 0$. If $f^{k}$ stands for the $k-$ th derivative then any polynomial of degree at most $k$ with leading coefficient small enough ; the answer is $f(z)=\sum_{j=0}^{k} c_j z^{j}$ with $|c_k| \leq \frac 1 {k!}$.