Inegalitate cu a,b,c,k>0 (OWN)

[Claudiu Mindrila]

Fie \( a,\ b,\ c,\ k\in\left(0,\ \infty\right \). Aratati ca

\( \left(\frac{a}{k}+\frac{k}{a}\right)^{2}+\left(\frac{b}{k}+\frac{k}{b}\right)^{2}+\left(\frac{c}{k}+\frac{k}{c}\right)^{2}\ge4\left(\frac{a+k}{b+k}+\frac{b+k}{c+k}+\frac{c+k}{a+k}\right) \).

Claudiu Mindrila, R. M. T. 4/2009

[Marius Mainea]

\( RHS\le\sum(a+k)(\frac{1}{b}+\frac{1}{k})=\sum\frac{a}{k}+\sum\frac{a}{b}+\sum\frac{k}{a}+3\le\frac{1}{2}\sum[(\frac{a}{k})^2+1]+\frac{1}{2}\sum[(\frac{a}{k})^2+(\frac{k}{b})^2]+\frac{1}{2}\sum[(\frac{k}{a})^2+1]+3=LHS \)