<s>[INST] Missing step in rearrangement Can someone explain the missing step in the following rearrangement ?
From $ \displaystyle a^2 + b^2 + \frac {a^2y}{x} + \frac {b^2x}{y} \geq (a+b)^2 $ to $ \displaystyle \frac{(a+b)^2}{x+y} \leq \frac{a^2}{x} + \frac{b^2}{y} $ ?
 [/INST] We have
$$ \left(\frac{a^2}{x} + \frac{b^2}{y}\right)(x+y)
=\frac{a^2}{x}x +\frac{a^2}{x}y+ \frac{b^2}{y}x +\frac{b^2}{y}y
=a^2 +\frac{a^2}{x}y+ \frac{b^2}{y}x +b^2y $$
and therefore
$$  \left(\frac{a^2}{x} + \frac{b^2}{y}\right)(x+y) = a^2 + b^2 + \frac {a^2y}{x} + \frac {b^2x}{y} \geq (a+b)^2. $$
And this gives, assuming $x+y>0$
$$ \frac{(a+b)^2}{x+y} \leq \frac{a^2}{x} + \frac{b^2}{y}.$$ 
If $x+y<0$, the inequality would be reversed!
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