Question Number 220159 by Nicholas666 last updated on 06/May/25
![for all x, y ∈ [0 , 1] ; prove that; (1/( (√(1 + x^4 )))) + (2/( (√(1 + y^4 )))) + (2/( (√(4 + (x + y)^4 )))) + ((2(√2))/( (√(2+ x^2 y^2 + y^3 )))) ≤ (2/( (√(1 + x^2 y^2 )))) + (2/(^4 (√(1 + x^5 + y^5 )))) + ln(e+((x^3 y+y^3 x)/(1 + xy))) + (1/((1+x+y)^3 ))](https://www.tinkutara.com/question/Q220159.png)
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{for}\:\mathrm{all}\:{x},\:{y}\:\in\:\left[\mathrm{0}\:,\:\mathrm{1}\right]\:;\:\mathrm{prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}\:+\:{x}^{\mathrm{4}} }}\:+\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{1}\:+\:{y}^{\mathrm{4}} }}\:+\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{4}\:+\:\left({x}\:+\:{y}\right)^{\mathrm{4}} }}\:+\:\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{2}+\:{x}^{\mathrm{2}} {y}^{\mathrm{2}} \:+\:{y}^{\mathrm{3}} }}\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\leqslant\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{1}\:+\:{x}^{\mathrm{2}} {y}^{\mathrm{2}} }}\:+\:\frac{\mathrm{2}}{\:^{\mathrm{4}} \sqrt{\mathrm{1}\:+\:{x}^{\mathrm{5}} \:+\:{y}^{\mathrm{5}} }}\:+\:\mathrm{ln}\left({e}+\frac{{x}^{\mathrm{3}} {y}+{y}^{\mathrm{3}} {x}}{\mathrm{1}\:+\:{xy}}\right)\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+{x}+{y}\right)^{\mathrm{3}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$
Commented by MrGaster last updated on 07/May/25
There may be mistakes in the topic, and the left side is greater than the right side at some points, which cannot prove the original inequality.