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Question Number 154267 by mathdanisur last updated on 16/Sep/21

if  x;y;z>0 and xy+yz+zx=3xyz  then:  (Σ_(cyc)  (1/(x+y)))(Σ_(cyc)  (1/(2x+y+z))) ≤ (9/8)

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\mathrm{and}\:\mathrm{xy}+\mathrm{yz}+\mathrm{zx}=\mathrm{3xyz}\:\:\mathrm{then}: \\ $$$$\left(\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\frac{\mathrm{1}}{\mathrm{x}+\mathrm{y}}\right)\left(\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\frac{\mathrm{1}}{\mathrm{2x}+\mathrm{y}+\mathrm{z}}\right)\:\leqslant\:\frac{\mathrm{9}}{\mathrm{8}} \\ $$

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