Prove that for all $\displaystyle x,y,z>0$
$\displaystyle \frac{x}{(x+y)(x+z)}+\frac{y}{(y+z)(y+x)}+\frac{z} {(z+x)(z+y)}\le \frac{9}{4(x+y+z)}
$
so we want to prove that: $\displaystyle 8(x+y+z)(xy+yz+zx) \leq 9(x+y)(y+z)(z+x),$ which after simplifying becomes: $\displaystyle 6xyz \leq x^2y+yz^2+x^2z +y^2z + xy^2+ xz^2 . \ \ \ \ (1)$
dividing both sides of (1) by $\displaystyle xyz$ gives us: $\displaystyle 6 \leq \frac{x}{z} + \frac{z}{x} + \frac{x}{y} + \frac{y}{x} + \frac{y}{z} + \frac{z}{y},$ which is clearly true by AM-GM, or (if you don't like AM-GM) because: $\displaystyle \forall t > 0 : \ t + \frac{1}{t} \geq 2. \ \ \Box$