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Question Number 155774 by mathdanisur last updated on 04/Oct/21
if  x;y>0  then prove that:  ((x^2 (1+xy)^2 +x^2 y^4 (1+x)^2 +(1+y)^2 )/(1 + xy + x^2 y + xy^2 )) ≥ 3xy
$$\mathrm{if}\:\:\mathrm{x};\mathrm{y}>\mathrm{0}\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{x}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{xy}\right)^{\mathrm{2}} +\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{4}} \left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} +\left(\mathrm{1}+\mathrm{y}\right)^{\mathrm{2}} }{\mathrm{1}\:+\:\mathrm{xy}\:+\:\mathrm{x}^{\mathrm{2}} \mathrm{y}\:+\:\mathrm{xy}^{\mathrm{2}} }\:\geqslant\:\mathrm{3xy} \\ $$

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