, 2014

Let $a$, $b$ and $c$ be three distinct positive reals. The logarithmic mean of $a$, $b$, $c$ is defined by
$L(a,b,c) = $\text{2 ( a ( a- b)( a-}$ c)} +
$\frac{b}{(\ln b-\ln c)(\ln b-\ln}$ a)} +
$\frac{c}{(\ln c-\ln a)(\ln c-\ln}$ b)} $\left. \right).\$$
Prove that $\sqrt[3]{abc} < L(a,b,c) < \frac{a+b+c}{3}$.
(5 pont)