if x;y;z>0 prove that ((x^3 +y^3 +z^3 )/(xyz)) ≥ 2((x/(y+z)) + (y/(z+x)) + (z/(x+y)))


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Question Number 147582 by mathdanisur last updated on 22/Jul/21

$i f x; y; z > 0 p r o v e t h a t$ $\frac{x^{3} + y^{3} + z^{3}}{x y z} ⩾ 2 (\frac{x}{y + z} + \frac{y}{z + x} + \frac{z}{x + y})$
	Answered by mitica last updated on 22/Jul/21

	$m_{h} ⩽ m_{a} \Rightarrow \frac{1}{a + b} ⩽ \frac{a + b}{4 a b}$ $2 Σ \frac{x}{y + z} ⩽ 2 Σ \frac{x (y + z)}{4 y z} = Σ \frac{x^{2} (y + z)}{2 x y z} =$ $\frac{1}{2 x y z} Σ x y (x + y) ⩽ \frac{1}{2 x y z} Σ (x^{3} + y^{3}) = \frac{2 (x^{3} + y^{3} + z^{3})}{2 x y z} = \frac{x^{3} + y^{3} + z^{3}}{x y z}$
	Commented bymathdanisur last updated on 22/Jul/21

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