Inegalitate conditionata cu produs

alex2008 · Post by **alex2008** » Mon Sep 14, 2009 3:55 pm

Fie \( x,y,z>0 \) astfel incat \( xyz=1 \). Sa se demonstreze ca :

\( \sum_{cyc}\sqrt{\frac{1}{1+x}}\le \frac{3\sqrt{2}}{2} \)

Marius Mainea · Post by **Marius Mainea** » Mon Sep 14, 2009 8:30 pm

Notam \( f(x,y,z)=\sum_{cyc}\sqrt{\frac{1}{1+x^2}} \) si inegalitatea este echivalenta cu

\( f(x,y,z)\le\frac{3\sqrt{2}}{2} \) cu \( xyz=1 \)

Putem presupune ca \( xy\le1 \).

Atunci \( f(x,y,z)\le f(\sqrt{xy},\sqrt{xy},z)\le \frac{3\sqrt{2}}{2} \)