<s>[INST] Combined AM GM QM inequality I came across this interesting inequality, and was looking for interesting proofs. $x,y,z \geq 0$
$$ 2\sqrt{\frac{x^{2}+y^{2}+z^{2}}{3}}+3\sqrt [3]{xyz}\leq 5\left(\frac{x+y+z}{3}\right) $$
Addendum. 
In general, when is 
$$ a\sqrt{\frac{x^{2}+y^{2}+z^{2}}{3}}+b\sqrt [3]{xyz}\leq (a+b)\left(\frac{x+y+z}{3}\right) $$
true?
 [/INST] Let $x+y+z=3u$, $xy+xz+yz=3v^2$ and $xyz=w^3$.
Hence, we need to prove that $f(w^3)\geq0$, where $f(w^3)=5u-3w-2\sqrt{3u^2-2v^2}$.
We see that $f$ is a decreasing function, 
which says that it's enough to prove our inequality for a maximal value of $w^3$,
which happens for equality case of two variables.
Indeed, $x$, $y$ and $z$ are non-negative roots of the equation
$$(X-x)(X-y)(X-z)=0$$ or
$$X^3-3uX^2+3v^2X-w^3=0$$ or
$$w^3=X^3-3uX^2+3v^2X$$ and we see that $w^3$ gets a maximal value, 
when a line $Y=w^3$ is a tangent line to the graph of $Y=X^3-3uX^2+3v^2X$,
which happens for equality case of two variables.
Since our inequality is homogeneous, we can assume $x=y=1$ and $z=t^3$.
Thus, it's enough to prove that:
$$\frac{5}{3}(t^3+2)-3t\geq2\sqrt{\frac{t^6+2}{3}}$$ or
$$(5t^3-9t+10)^2\geq12(t^6+2)$$ or
$$(t-1)^2(13t^4+26t^3-51t^2-28t+76)\geq0,$$
which is obvious.
Done!
By the same way we can get a best values of $a$ and $b$.
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