Question Number 220160 by Nicholas666 last updated on 06/May/25
![for all x , y [0 , 1] ; prove that; [ (((x^3 + y^3 + 𝛇(3)))^(1/(3 )) /(1 + e^(−x^2 y^2 ) )) + (((x^4 + 𝚪(y+1)))^(1/(4 )) /((1 + y^2 )^(1/3) )) + ((ln(1 + x^5 + y^5 ))/( (√(1 + x^2 + y^2 )))) + Li_2 (xy) + ((√(x^6 + y^6 +1 ))/((1 + x^3 y^3 )^(1/2) )) ≤ (e^(xy) /(1 + x + y )) + ((ln (1 + x^2 + y^2 ) ))^(1/(3 )) + ((2𝛇(2))/( (√(1 + x^2 y^2 )))) + ((x^8 + y^8 + 1))^(1/(4 )) ]](https://www.tinkutara.com/question/Q220160.png)
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{for}\:\mathrm{all}\:{x}\:,\:{y}\:\left[\mathrm{0}\:,\:\mathrm{1}\right]\:;\:\mathrm{prove}\:\mathrm{that}; \\ $$$$\:\:\left[\:\frac{\sqrt[{\mathrm{3}\:\:}]{\boldsymbol{{x}}^{\mathrm{3}} \:+\:\boldsymbol{{y}}^{\mathrm{3}} \:+\:\boldsymbol{\zeta}\left(\mathrm{3}\right)}}{\mathrm{1}\:+\:\boldsymbol{{e}}^{−\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} } \:}\:+\:\frac{\sqrt[{\mathrm{4}\:\:}]{\boldsymbol{{x}}^{\mathrm{4}} +\:\boldsymbol{\Gamma}\left(\boldsymbol{{y}}+\mathrm{1}\right)}}{\left(\mathrm{1}\:+\:\boldsymbol{{y}}^{\mathrm{2}} \right)^{\mathrm{1}/\mathrm{3}} }\:+\:\frac{\boldsymbol{\mathrm{ln}}\left(\mathrm{1}\:+\:\boldsymbol{{x}}^{\mathrm{5}} \:+\:\boldsymbol{{y}}^{\mathrm{5}} \right)}{\:\sqrt{\mathrm{1}\:+\:\boldsymbol{{x}}^{\mathrm{2}} \:+\:\boldsymbol{{y}}^{\mathrm{2}} }}\:\:\:\:\:\:\:\:\right. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:+\:\boldsymbol{\mathrm{Li}}_{\mathrm{2}} \left(\boldsymbol{{xy}}\right)\:+\:\frac{\sqrt{\boldsymbol{{x}}^{\mathrm{6}} \:+\:\boldsymbol{{y}}^{\mathrm{6}} \:+\mathrm{1}\:}}{\left(\mathrm{1}\:+\:\boldsymbol{{x}}^{\mathrm{3}} \boldsymbol{{y}}^{\mathrm{3}} \right)^{\mathrm{1}/\mathrm{2}} }\: \\ $$$$\left.\:\:\:\:\:\:\leqslant\:\frac{\boldsymbol{{e}}^{\boldsymbol{{xy}}} }{\mathrm{1}\:+\:\boldsymbol{{x}}\:+\:\boldsymbol{{y}}\:}\:+\:\sqrt[{\mathrm{3}\:\:}]{\boldsymbol{\mathrm{ln}}\:\left(\mathrm{1}\:+\:\boldsymbol{{x}}^{\mathrm{2}} \:+\:\boldsymbol{{y}}^{\mathrm{2}} \right)\:}\:+\:\frac{\mathrm{2}\boldsymbol{\zeta}\left(\mathrm{2}\right)}{\:\sqrt{\mathrm{1}\:+\:\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}^{\mathrm{2}} }}\:+\:\sqrt[{\mathrm{4}\:\:}]{\boldsymbol{{x}}^{\mathrm{8}} \:+\:\boldsymbol{{y}}^{\mathrm{8}} \:+\:\mathrm{1}}\:\:\:\:\:\:\:\:\right]\:\: \\ $$$$ \\ $$$$\:\: \\ $$