Inegalitate in trei variabile subunitare

[alex2008]

Fie \( x,y,z\in (0,1) \) . Sa se demonstreze ca :

\( \frac{3 + xy}{1 - z} + \frac{3 + yz}{1 - x} + \frac{3 + zx}{1 - y}\ge 4\left(\frac{x + y}{1 - xy} + \frac{y + z}{1 - yz} + \frac{z + x}{1 - zx}\right) \)

Virgil Nicula

[Marius Mainea]

Avem

\( \frac{1}{1-xy}\le\frac{1}{1-\frac{x^2+y^2}{2}}=\frac{2}{(1-x^2)+(1-y^2)}\le\frac{\frac{1}{1-x^2}+\frac{1}{1-y^2}}{2} \)

Asadar \( RHS\le\sum 2\frac{2x+y+z}{1-x^2}\le\sum \frac{3+yz}{1-x}=LHS \) deoarece

\( 2(2x+y+z)\le (1+x)(3+yz) \) (demonstrati)