Math people:

I would like a reference for the following fact (?), which I proved myself (I am 99% sure the proof is valid) but which has probably been done before. My proof was a little messy. If no one can supply a reference, I would appreciate an elegant proof. Here is what I proved:

Let $\Omega$ be an open subset of $\mathbf{R}^n$, $\delta >0$, and $K >0$. Then there exists $\kappa = \kappa(\delta, K, \Omega)>0$ with the following property: if $f , g \in C^\infty(\Omega)$ with $\inf_\Omega f \geq 0$, $\inf_\Omega g \geq 0$, $\int_\Omega f\,dx \leq K$, and $\int_\Omega g\,dx \geq \delta$, then

$$ \int_\Omega \sqrt{f^2+g^2} - f \,dx > \kappa.$$

Please note that there are no assumptions on $\sup_\Omega f$: if there were, the proof would be easier. My proof did not use the smoothness of $f$ or $g$, and my $\kappa$ did not involve $\Omega$ at all. I would be OK with a $\kappa$ that did involve $\Omega$. I do not care how sharp the inequality is. I only need the result for $\Omega$ being a two-dimensional rectangle.

I need this as part of a larger proof and it would be better to cite this fact, if it has been done before, than to include it in the paper.

  • $\begingroup$ For the two-dimensional case, why not let $h=f+ig$ (viewed as a map $h:\mathbb{C} \to \mathbb{C}$) and consider the integral $\int_\Omega (\vert h \vert - \mathrm{Re}(h))$? This might be easier to find references for. $\endgroup$ – awwalker May 16 '13 at 18:16

One way of proving it is to write $$\int\frac{g^2}{\sqrt{f^2+g^2}+f}$$ and use CS to end up with $$\int\frac{g^2}{\sqrt{f^2+g^2}+f}\int\sqrt{f^2+g^2}+f\ge\left(\int g\right)^2.$$ Now $\int\sqrt{f^2+g^2}+f\le \int f+g+f\le 2K+\int g.$ So our integral bounded below by $$\frac{\left(\int g\right)^2}{K+\int g}$$ which can be easily estimated via $\delta.$

  • $\begingroup$ That is really slick. It is hard to imagine a simpler solution. I'll wait a few days and see if someone can find a reference. $\endgroup$ – Stefan Smith May 16 '13 at 19:13
  • $\begingroup$ I am writing a paper in which I will use your proof in a lemma. I would like to give you some credit. If you would rather remain anonymous I understand. $\endgroup$ – Stefan Smith Jun 12 '13 at 23:50
  • $\begingroup$ @Stefan: I do not mind giving you my real name. Drop me an email at mathleshik@gmai.com and I will reply. $\endgroup$ – leshik Jun 13 '13 at 14:43

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