A question about $L^p$ integral of an entire function on $\mathbb{C}$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T10:59:44Z http://mathoverflow.net/feeds/question/117662 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117662/a-question-about-lp-integral-of-an-entire-function-on-mathbbc A question about $L^p$ integral of an entire function on $\mathbb{C}$ woodbass 2012-12-30T19:47:27Z 2013-04-26T17:21:53Z <p>Question: Suppose that $f$ is an entire function (i.e. analytic in $\mathbb{C}$), and satisfies the condition $\iint_{\mathbb{C}}|f|^p dxdy&lt;\infty$ for some $p\in (0,1)$. I guess that $f\equiv 0$ but I do not know how to prove it.</p> <p>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.</p> <p>EDIT. Thank Alexandre Eremenko for his answer. I also want to know the solution to the following harmonic version of question.</p> <p>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 $\iint_{\mathbb{C}}|f|^p dxdy&lt;\infty$ for some $p\in (0,1).$ I believe that $f\equiv 0$.</p> http://mathoverflow.net/questions/117662/a-question-about-lp-integral-of-an-entire-function-on-mathbbc/117667#117667 Answer by Alexandre Eremenko for A question about $L^p$ integral of an entire function on $\mathbb{C}$ Alexandre Eremenko 2012-12-30T21:02:09Z 2013-01-01T15:54:02Z <p>Yes, this is so. And much stronger statements are available: See the beautiful survey paper</p> <p>MR2567024 Rashkovskii, Alexander Classical and new loglog-theorems. Expo. Math. 27 (2009), no. 4, 271–287.</p> <p>also available on the arxiv</p> <p>All results in this paper are actually for subharmonic functions, so this settles the questrion for both analytic and harmonic functions.</p> http://mathoverflow.net/questions/117662/a-question-about-lp-integral-of-an-entire-function-on-mathbbc/128455#128455 Answer by abas for A question about $L^p$ integral of an entire function on $\mathbb{C}$ abas 2013-04-23T08:53:55Z 2013-04-23T08:53:55Z <p>I am a student and wondering which Theorem exactly answers the first question</p> http://mathoverflow.net/questions/117662/a-question-about-lp-integral-of-an-entire-function-on-mathbbc/128843#128843 Answer by pooper for A question about $L^p$ integral of an entire function on $\mathbb{C}$ pooper 2013-04-26T17:21:53Z 2013-04-26T17:21:53Z <p>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 $f \equiv 0$.</p> <p>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.</p>