Question: Suppose that $f$ is an entire function (i.e. analytic in $\mathbb{C}$), and satisfies the condition $\iint\_{\mathbb{C}}|f|^p dxdy<\infty$ for some $p\in (0,1)$. I guess that $f\equiv 0$ but I do not know how to prove it.


Note. If $p\in[1,\infty)$, it is easy to prove that $f\equiv 0$. In the settting $p\in (0,1)$, one should deal with the integral of an entire function near the essential singularity point $\infty$ carefully.


EDIT. Thank Alexandre Eremenko for his answer. I also want to know the solution to the following harmonic version of question.


Question (H): Suppose that $f$ is a harmonic function ($i.e. \Delta f=0$, $f$ may be complex-harmonic in $\mathbb{C}$), and satisfies the condition $\iint\_{\mathbb{C}}|f|^p dxdy<\infty$ for some $p\in (0,1).$ I believe that $f\equiv 0$.



A question about $L^p$ integral of an entire function on $\mathbb{C}$
Yes, this is so. And much stronger statements are available:
See the beautiful survey paper


MR2567024 Rashkovskii, Alexander Classical and new loglog-theorems. 
Expo. Math. 27 (2009), no. 4, 271–287.


also available on the arxiv


All results in this paper are actually for subharmonic functions, so this settles
the questrion for both analytic and harmonic functions.


