looking for reference for integral inequality 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.

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 $\delta.$
