Monotonicity of Average Power Means

Abstract

A new numerical inequality for average power means is presented. Let $\alpha ,\beta \in \left[ { - \infty + \infty } \right]$ and let $a = \left( {a_k } \right)_{k \geqslant 1}$ be a sequence of positive numbers. Consider the operator $M_\alpha \left( a \right) = \left\{ {\left( {\frac{{a_1^\alpha + a_2^\alpha + \cdot \cdot \cdot + a_k^\alpha }}{\kappa }} \right)^{\frac{1}{\alpha }} } \right\}_{k \geqslant 1}$. We denote by $M_\beta {\text{ o }}M_\alpha$ the superposition of these operators. The following assertion is proved: if $\alpha < \beta , then{\text{ }}M_\beta {\text{ o }}M_\alpha \left( a \right) \leqslant M_\alpha {\text{ o }}M_\beta \left( a \right)$. Bibliography: 2 titles.