Inequality for integrals of probability densities

Question

Given three univariate probability densities, $f(x)$, $g(x)$, and $h(x)$, I would like to show that $\int_{supp(f)\cap supp(g)}\frac{fg}{f+g}dx+\int_{supp(g)\cap supp(h)}\frac{gh}{g+h}dx-\int_{supp(f)\cap supp(h)}\frac{fh}{f+h}dx\leq\frac{1}{2}$

where $supp$ is the support of a density. I am sure that this is quite straightforward, but I am a bit stuck as many of the obvious inequalities are too crude.

Answer 1

Say $f$ is uniform in $[0,1/2]$, $g$ is uniform in $[0,1]$, and $h$ is uniform in $[1/2,1]$. Then you get

$$ \int_0^{1/2} \frac{2\cdot 1}{2+1} + \int_{1/2}^{1} \frac{1\cdot 2}{1+2} dx -\int_{\emptyset} dx = \frac 1 3 + \frac 1 3 - 0 = \frac 2 3 $$