KöMaL 2014 A623

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