<s>[INST] Showing that $ \left( \frac{a+2b}{a+2c}\right)^3+\left( \frac{b+2c}{b+2a}\right)^3+\left( \frac{c+2a}{c+2b}\right)^3 \geq3 $ I would like to show that:
$$ \left( \frac{a+2b}{a+2c}\right)^3+\left( \frac{b+2c}{b+2a}\right)^3+\left( \frac{c+2a}{c+2b}\right)^3 \geq3 $$
Using AM-GM inequality:
$$ \left( \frac{a+2b}{a+2c}\right)^3+\left( \frac{b+2c}{b+2a}\right)^3+\left( \frac{c+2a}{c+2b}\right)^3 \geq3\frac{(a+2b)(b+2c)(c+2a)}{(a+2c)(b+2a)(c+2b)} $$
It suffices to show that:
$$ \frac{(a+2b)(b+2c)(c+2a)}{(a+2c)(b+2a)(c+2b)} \geq1 $$
$$ \Longleftrightarrow ab^2+bc^2+ca^2-(a^2b+b^2c+c^2a)\geq0$$
$$ \Longleftrightarrow x+y+z\geq \frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$
$$ xyz=1$$
($x=\frac{a}{b}$, $y=\frac{b}{c}$, $z=\frac{c}{a}$)
How can I prove this last inequality?
Is there any simpler proof?
 [/INST] I will prove that this is true for $a,b,c>0$. If they're allowed to be non-positive, there are counterexamples; e.g., $(a,b,c)=(1,0,-1)$.
Since this expression is homogeneous in $a,b,c$, we may assume without loss of generality that $a+b+c=1$. Then setting $x=c-b$, $y=a-c$, $z=b-a$ yields:
$$
\left(\frac{a+2b}{a+2c}\right)^3+\left(\frac{b+2c}{b+2a}\right)^3+\left(\frac{c+2a}{c+2b}\right)^3=\left(\frac{1-x}{1+x}\right)^3+\left(\frac{1-y}{1+y}\right)^3+\left(\frac{1-z}{1+z}\right)^3 \, .
$$
But if $a,b,c$ are positive and $a+b+c=1$, then $0<a,b,c<1$, from which it follows that $-1<x,y,z<1$. As the function $f(t)=\left(\frac{1-t}{1+t}\right)^3$ is concave upward on $(-1,1)$, we have:
$$
\left(\frac{1-x}{1+x}\right)^3+\left(\frac{1-y}{1+y}\right)^3+\left(\frac{1-z}{1+z}\right)^3=f(x)+f(y)+f(z) \geq 3f\left(\frac{x+y+z}{3}\right)=3f(0)=3 \, .
$$
This completes the proof.
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