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If-x-y-z-gt-0-then-prove-that-x-x-2-yz-y-y-2-zx-z-z-2-xy-x-2-y-2-z-2-2xyz-




Question Number 154015 by mathdanisur last updated on 13/Sep/21
If  x;y;z>0  then prove that:  (x/(x^2 +yz)) + (y/(y^2 +zx)) + (z/(z^2 +xy)) ≤ ((x^2 +y^2 +z^2 )/(2xyz))
$$\mathrm{If}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{x}}{\mathrm{x}^{\mathrm{2}} +\mathrm{yz}}\:+\:\frac{\mathrm{y}}{\mathrm{y}^{\mathrm{2}} +\mathrm{zx}}\:+\:\frac{\mathrm{z}}{\mathrm{z}^{\mathrm{2}} +\mathrm{xy}}\:\leqslant\:\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} }{\mathrm{2xyz}} \\ $$

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