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 $f\equiv 0$.

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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: 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 $p\in (0,1)$. 0,1).$I believe that$f\equiv 0$. 3 added 367 characters in body Question: Suppose that$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$carefully. EDIT. Thank Alexandre Eremenko for his answer. I also want to know the solution to the following harmonic version of question. Question: Suppose that$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\$.

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