Scaled evolution of the reversed power mean inequalities

Abstract

Motivated by the Hanin inequality, in this paper we study a class of  reversed power mean inequalities that does not depend on a weight function.
We first give the reverse of the basic power mean inequality describing the monotonic behavior of means. Then, we establish the scaled, i.e. two-parametric versions of the obtained inequalities. By scaling, we develop a new method for improving the starting reversed power mean inequalities. More precisely, we impose conditions under which the scaled inequality is sharper than the corresponding original power mean inequality. Some particular examples are also discussed. Finally, our results are compared with some previously known from the literature.