Generalization and sharpness of the power means inequality and their applications

In this paper, we generalize and sharpen the power means inequality by using the theory of majorization and the analytic techniques. Our results unify some optimal versions of the power means inequality. As application, a well-known conjectured inequality proposed by Janous et al. is proven. Furthermore, these results are used for studying a class of geometric inequalities for simplex, from which, some interesting inequalities including the refined Euler inequality and the reversed Finsler-Hadwiger type inequality are obtained.