A simple proof of inequalities related to means

Abstract

The purpose of this paper is to give a fairly elementary method to prove that the function $u(\alpha)=[\frac{x^{\alpha}-y^{\alpha}}{\alpha(x-y)}]^{\frac{1}{\alpha -1}},$ $\alpha \neq 0,1, u(0)=I, u(1)=L, $ is strictly increasing, and to give a simple proof of the inequalities $x