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KöMaL problem A, 2014

Algebra Difficulty 6.0 National Olympiad Prove it Hungary

Let aa, bb and cc be three distinct positive reals. The logarithmic mean of aa, bb, cc is defined by
L(a,b,c)=2(a(ln⁡a−ln⁡b)(ln⁡a−ln⁡c)+b(ln⁡b−ln⁡c)(ln⁡b−ln⁡a)+c(ln⁡c−ln⁡a)(ln⁡c−ln⁡b)).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 abc3<L(a,b,c)<a+b+c3\sqrt[3]{abc} < L(a,b,c) < \frac{a+b+c}{3}.
(5 pont)

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