<s>[INST] Find the Minimum value of this expression Let $x,y,z>0$ such that: $x+y+z=1$. Find the minimum value of this expression.
$P=(\frac{x+1}{x})^3.\frac{z^2}{z^2+1}+(\frac{y+1}{y})^3.\frac{x^2}{x^2+1}+(\frac{z+1}{z})^3.\frac{y^2}{y^2+1}$
 [/INST] Using rearrangement inequality:
$$P=\sum_{cyc} \left[\left(\frac{x+1}{x}\right)^3 \frac{z^2}{z^2+1}\right] \ge \sum \left[\left(\frac{x+1}{x}\right)^3 \frac{x^2}{x^2+1}\right] = \sum \frac {(1+x)^3}{x \, (1+x^2)}$$
Noting $f(x) = \dfrac{(1+x)^3}{x \, (1+x^2)}$ is convex in $(0, 1)$, by Jensen's inequality:
$$\frac{P}{3} \ge \frac {(1+\frac13)^3}{\frac13 \, (1+\frac19)} = \frac{32}5$$
So the minimum of $P$ is $\frac{96}5$, achieved when $x=y=z=\frac13$.
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