Inegalitate

[Al3xx]

\( Fie\ x,y,z>0. \)

\( Sa\ se\ demonstreze\ ca\ : \)

\( \frac{xy}{xy+x+y}+\frac{yz}{yz+y+z}+\frac{zx}{zx+z+x} \le\frac{2}{3}+\frac{x^2+y^2+z^2}{9} \)

[Marius Mainea]

Inegalitatea este echivalenta cu

\( \sum {\frac{x+y}{x+y+xy}}+\frac{x^2+y^2+z^2}{9}\ge \frac{7}{3} \).

Insa \( LHS\ge \sum {\frac{x+y}{x+y+xy}}+\frac{xy+yz+zx}{9}\ge RHS \),

deoarece \( \frac{x+y}{x+y+xy}+\frac{xy}{9}\ge \frac{7}{9} \) si analoagele.