## A question about $L^p$ integral of an entire function on $\mathbb{C}$

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 $f\equiv 0$.

-

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.

-
 What about the answer to the harmonic version of question in my EDIT? – woodbass Dec 31 at 1:01 Rashkovski's paper also discusses the harmonic, and more generally, subharmonic versions. – Alexandre Eremenko Feb 8 at 14:40

In the case of $f$ entire a much more straightforward solution is availabe. First if $f$ is entire, then $|f|^{p}$ is subharmonic. Secondly, by subharmonicity, for any $z_0$, we have $$|f(z_0)|^{p} \leq \frac{1}{\text{meas}({D(z_0,R)})} \iint_{D(z_0,R)} |f(z)|^{p} dz \leq \frac{C}{\text{meas}(D(z_0,R))}$$ where $D(z_0,R)$ is a disc of radius $R$ around $z_0$ and $C = \iint_{\mathbb{C}} |f(z)|^{p} dz$. Taking $R \rightarrow \infty$ in the above inequality we obtain $f(z_0) = 0$. Since $z_0$ was arbitrary $f \equiv 0$.

If the implications $f$ harmonic implies $|f|^p$ subharmonic'' is correct then this would also answer the second part of the question. I haven't checked if it's true, but it seems correct at least in the case $p = 1$ (which leaves me hoping that it's correct for all $p > 0$). If anybody could check and post the answer in the comments I will be indebted.

-
 You may want to look at the Wikipedia article on subharmonic functions. – S. Carnahan♦ May 14 at 13:58

I am a student and wondering which Theorem exactly answers the first question

-
What exactly did you not understand in my previous comment? And what exactly does not fit your question in Alexandre's answer? – Loïc Teyssier Apr 23 at 9:57
I have not yet understood from which theorem in the refereed paper the answer for the first question follows. Just help me understand it:) – abas Apr 23 at 10:04
I haven't read the paper, but I guess Alexandre's pointer was correct. In that case I surmise that the answer is :"that's for you to find out". – Loïc Teyssier Apr 23 at 10:21