Find all entire functions with $\int_{\Bbb C} |f(z)|\, dz= 1$ I know that an entire function with bounded $L^1$ norm is identically $0$, but I do not know how to attack this problem. Does this contradict the fact I stated about entire functions with bounded $L^1$ norm? 

There are no such functions.
For each $w \in \mathbb C$, we have
$$
|f(w)| \le \int_{\Bbb C} |f(z)|\, dz= 1
$$
Therefore, $f$ is bounded. Since $f$ is entire, $f$ is constant, by Liouville's theorem.
But then we cannot have $\int_{\Bbb C} |f(z)|\, dz= 1$.
