I stumbled upon the following statement
Let $p>2$ and let $f$ be an entire function (i.e. holomorphic on the whole $\mathbb{C}$), and assume that $f'-1$ is in $L^p(\mathbb{C})$. Then $f'\equiv 1$
To me, it looks like the author is suggesting a statement of the form:
There exists no integrable, non-zero, entire function.
(as I presume that the $p>2$ doesn't play any role here... Or maybe it does?)
But I cannot seem to find/prove this. The only thing that has come to my mind, which could be helpful, is Liouville's theorem, but this doesn't seem to be enough, as I can think of examples of $C^\infty$ functions that are unbounded and yet integrable. Or maybe Green's theorem for the complex plane?
I must admit, I am not very confident with the subject at the moment, as I had my Complex Analysis class more than two years ago.
Does someone have a solution or a reference for this?
Thank you in advance