Applying Liouville's theorem

Question

Let $f : \mathbb C \rightarrow \mathbb C$ be an entire function such that $2f(1) = f(0)$. What can you say about $f$?

Question is asked under Liouville's theorem, so it must be some direct application of it, but I can't think of a suitable way

I can think of one such entire function, namely $f(z) = 2-z$ and multiples of it.

Answer 1

You can't say much. There are lots of such functions. In fact, you can specify arbitrarily the values of $f$ at any sequence of points of $\mathbb C$ with no finite limit point, and there will be an entire function with those values at those points.