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If-0-lt-a-b-lt-1-then-a-b-a-b-a-b-1-xyz-1-xyz-3-dxdydz-a-b-1-x-3-1-x-3-dx-3-




Question Number 153808 by mathdanisur last updated on 10/Sep/21
If  0<a≤b<1  then:  ∫_( a) ^( b) ∫_( a) ^( b) ∫_a ^( b) (((1 - xyz)/(1 + xyz)))^3 dxdydz ≥ (∫_a ^( b) ((1 - x^3 )/(1 + x^3 )) dx)^3
$$\mathrm{If}\:\:\mathrm{0}<\mathrm{a}\leqslant\mathrm{b}<\mathrm{1}\:\:\mathrm{then}: \\ $$$$\underset{\:\boldsymbol{\mathrm{a}}} {\overset{\:\boldsymbol{\mathrm{b}}} {\int}}\underset{\:\boldsymbol{\mathrm{a}}} {\overset{\:\boldsymbol{\mathrm{b}}} {\int}}\underset{\boldsymbol{\mathrm{a}}} {\overset{\:\boldsymbol{\mathrm{b}}} {\int}}\left(\frac{\mathrm{1}\:-\:\mathrm{xyz}}{\mathrm{1}\:+\:\mathrm{xyz}}\right)^{\mathrm{3}} \mathrm{dxdydz}\:\geqslant\:\left(\underset{\boldsymbol{\mathrm{a}}} {\overset{\:\boldsymbol{\mathrm{b}}} {\int}}\frac{\mathrm{1}\:-\:\mathrm{x}^{\mathrm{3}} }{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{3}} }\:\mathrm{dx}\right)^{\mathrm{3}} \\ $$

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