if x;y;z>1 then: (√((((x−1)(y−1)(z−1))/((x+1)(y+1)(z+1))) ))


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Question Number 147892 by mathdanisur last updated on 24/Jul/21

$i f x; y; z > 1 t h e n :$ $\sqrt{\frac{(x - 1) (y - 1) (z - 1)}{(x + 1) (y + 1) (z + 1)}} < \frac{x y z}{8}$
	Answered by mindispower last updated on 24/Jul/21

	$\frac{x - 1}{x + 1} < \frac{x^{2}}{4} . . . ? \forall x > 1$ $\Leftrightarrow 4 (x - 1) < x^{2} (x + 1)$ $\Leftrightarrow x^{3} + x^{2} + 4 - 4 x > 0$ $\Leftrightarrow x (x^{2} + 1) + 4 - 4 x > 0$ $x^{2} + 1 ⩾ 2 x \Rightarrow x (x^{2} + 1) + 4 - 4 x > 2 x^{2} - 4 x + 4 = 2 {(x - 1)}^{2} + 2 > 0$ $\Rightarrow \forall x > 1 \frac{x - 1}{x + 1} < \frac{x^{2}}{4}, \frac{y - 1}{y + 1} < \frac{y^{2}}{4}, \frac{z - 1}{z + 1} < \frac{z^{2}}{4}$ $\Rightarrow \frac{(x - 1) (y - 1) (z - 1)}{(x + 1) (y + 1) (z + 1)} < \frac{x^{2} y^{2} z^{2}}{64}$ $\Rightarrow \sqrt{\frac{(x - 1) (y - 1) (z - 1)}{(x + 1) (y + 1) (z + 1}} < \frac{x y z}{8}$
	Commented by mathdanisur last updated on 24/Jul/21

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