Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have trouble with the following small exercise in complex analysis.
Let $R=Ann(0;0,1)=\{z\in\mathbb{C}\,|\,0<|z|<1\}$, and $f$ is an analytic function on $R$. For a fixed
$p>0$, if $f$ satisfies
$$\iint\limits_R |f(x+iy)|^p dxdy < \infty$$, what can we say about the type of singularity $f$ has at $0$?

This exercise is taken from Conway,John B. Functions of One Complex Variable. v1 (GTM 11) p.112.
I have no trouble to deal with the case when $p\ge 2$. Since when $p=2$, one can use Parseval's identity
to derive that the singular part of the Laurent series of $f$ is $0$. But when $0<p<2$, it seems not easy
to reduce to the case when $p=2$ simply by considering $z^n f(z)$ for a suitable $n$ as I had expected to
work. But it doesn't work for me.

I am guessing the answer is that $0$ is in fact a pole of order $m=\lceil\frac{2}{p}-1\rceil$ by considering the function $f(z)=z^{-m}$. Am I right ? Can anyone give me some hints ? Any help is appreciated.

share|improve this question

migrated from mathoverflow.net Jan 28 at 0:27

This question came from our site for professional mathematicians.

1 Answer 1

up vote 1 down vote accepted

Function $|f|^p$ is subharmonic for each $p>0$. For small $r>0$ solve the Dirichlet problem for the exterior of the disc $|z|<r$ with boundary values $|f(re^{i\theta})|^p$. This can be done explicitly by a version of the Poisson formula. Let $u(z)$ be this solution. Using subharmonicity, from the explicit formula, you obtain an estimate of the form $$|f(2re^{i\theta})|\leq u(re^{i\theta})+C_1\leq C\int_0^{2\pi}|f(re^{i\theta})|^pd\theta+C_1=:Cm_p(r)+C_1,$$ where $C$ is an absolute constant and $C_1$ depends on $f$ only, and where $m_p$ is the $L^p$ "norm". This does the business, by passing from $L_p$ to the uniform estimate. So under your condition, the function has at most a pole, and the order of the pole can be easily determined, as you did.

Remark. I know from experience that some problems in Conway can be much harder than expected in an undergraduate course. So a teacher making home assignment should be careful. In one extreme case, he himself could not provide a solution, when asked:-)

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.