MathDB

31

Part of 1974 IMO Longlists

Problems(1)

n variables inequality with α, λ [ILL 1974]

Source:

1/2/2011
Let yα=∑i=1nxiαy^{\alpha}=\sum_{i=1}^n x_i^{\alpha} where α≠0,y>0,xi>0\alpha \neq 0, y > 0, x_i > 0 are real numbers, and let λ≠α\lambda \neq \alpha be a real number. Prove that yλ>∑i=1nxiλy^{\lambda} > \sum_{i=1}^n x_i^{\lambda} if α(λ−α)>0,\alpha (\lambda - \alpha) > 0, and yλ<∑i=1nxiλy^{\lambda} < \sum_{i=1}^n x_i^{\lambda} if α(λ−α)<0.\alpha (\lambda - \alpha) < 0.
inequalitiesinequalities proposed