<s>[INST] Unknown Inequality $$ \left( \sqrt{3}\sqrt{y(x+y+z)}+\sqrt{xz}\right)\left( \sqrt{3}\sqrt{x(x+y+z)}+\sqrt{yz}\right)\left( \sqrt{3}\sqrt{z(x+y+z)}+\sqrt{xy}\right) \leq 8(y+x)(x+z)(y+z)$$
I can prove this inequality, but i need know if this inequaliy is known...
 [/INST] Here is a nice and simple proof (for $x,y,z\ge 0$ of course):
By Cauchy-Schwarz inequality:
$$\left( \sqrt{3}\sqrt{y(x+y+z)}+\sqrt{xz}\right)^2 \le (3+1)\big[(y(x+y+z) + xz\big] = 4(x+y)(y+z).$$
Similarly:
$$\left( \sqrt{3}\sqrt{x(x+y+z)}+\sqrt{yz}\right)^2 \le 4(x+z)(x+y)$$
$$\left( \sqrt{3}\sqrt{z(x+y+z)}+\sqrt{xy}\right)^2 \le 4(x+z)(y+z).$$
Taking the product of the above three inequalities we get the desired inequality.
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