Sa se arate ca :
\( \sqrt{x(y+1)}+\sqrt{y(z+1)}+\sqrt{z(x+1)}\le \frac{3}{2}\sqrt{(x+1)(y+1)(z+1)} \)
oricare ar fi x,y,z numere reale pozitive.
Gheorghe Szollosy,Gazeta Matematica
Inegalitate neomogena
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Marius Mainea
- Gauss
- Posts: 1077
- Joined: Mon May 26, 2008 2:12 pm
- Location: Gaesti (Dambovita)
Impartim prin radicalul mare din dreapta si obtinem
\( \sum_{cyc}\sqrt{\frac{x}{(x+1)(z+1)}}\le \frac{3}{2} \)
Acum folosim \( ab\le \frac{1}{2}\cdot (a^2+b^2) \) si avem :
\( S\le \frac{1}{2}\cdot (\sum_{cyc}\frac{x}{x+1}+\sum_{cyc}\frac{1}
{x+1})=\frac{1}{2}\cdot (\sum_{cyc}\frac{x}{x+1})=\frac{3}{2} \)
\( \sum_{cyc}\sqrt{\frac{x}{(x+1)(z+1)}}\le \frac{3}{2} \)
Acum folosim \( ab\le \frac{1}{2}\cdot (a^2+b^2) \) si avem :
\( S\le \frac{1}{2}\cdot (\sum_{cyc}\frac{x}{x+1}+\sum_{cyc}\frac{1}
{x+1})=\frac{1}{2}\cdot (\sum_{cyc}\frac{x}{x+1})=\frac{3}{2} \)
. A snake that slithers on the ground can only dream of flying through the air.