Inegalitate conditionata cu produs

[alex2008]

Fie \( a,b,c>0 \) astfel incat \( abc=1 \). Sa se arate ca :

\( \frac{1}{\sqrt{b+\frac{1}{a}+\frac{1}{2}}}+\frac{1}{\sqrt{c+\frac{1}{b}+\frac{1}{2}}}+\frac{1}{\sqrt{a+\frac{1}{c}+\frac{1}{2}}}\ge \sqrt{2} \)

[Marius Mainea]

Aplicam GM-HM,

\( LHS= \sum {\sqrt{\frac{2a}{2ab+2+a}}}\ge\sqrt{2}\sum {\frac{2}{1+\frac{2ab+2+a}{a}}=\sqrt{2}\sum {\frac{a}{ab+1+a}}=\sqrt{2}\cdot 1=RHS \)