The monotonicity of average power means

A new numerical inequality for average power means is presented. Let $\alpha,\beta\in[-\infty,+\infty]$ and let $a=(a_k)_{k\ge1}$ be a sequence of positive numbers. Consider the operator $M_{\alpha}(a)=\biggl\{\biggl(\dfrac{a_1^{\alpha}+a_2^{\alpha}+\ldots+a_k^{\alpha}}k\biggr)^\frac1{\alpha}\biggr\}_{k\ge1}$. We denote by $M_{\beta}\circ M_{\alpha}$ the superposition of these operators. The following assertion is proved: if $\alpha<\beta$, then $M_{\beta}\circ M_{\alpha}(a)\le M_{\alpha}\circ M_{\beta}(a)$.