From the Ingham--Jessen property to mixed-mean inequalities

Abstract

For every symmetric mean $\mathcal{M}:{\bigcup }_{n=1}^{\infty }{I}^n\to I$ (where $I$ an interval) and a nonzero function $W:\{1,\dots ,n\}\to \mathbb{N}\cup \{0\}$, define an $n$-variable mean by $${\mathcal{M}}_W(x):=\mathcal{M}(\underset{W(1)\text{-times}}{\underbrace{x_1,\dots ,x_1}},\dots ,\underset{W(n)\text{-times}}{\underbrace{x_n,\dots ,x_n}})\text{ for }x=(x_1,\dots ,x_n)\in I^n.$$ Given two symmetric means $\mathcal{M},\mathcal{N}:{\bigcup }_{n=1}^{\infty }{I}^n\to I$ satisfying the so-called Ingham--Jessen inequality and some nonzero functions $F_1,\dots ,F_k$, $G_1,\dots ,G_l:\{1,\dots ,n\}\to \mathbb{N}\cup \{0\}$, we establish sufficient conditions for inequalities of the form $$\mathcal{N}({\mathcal{M}}_{F_1}(x),\dots ,{\mathcal{M}}_{F_k}(x))\le \mathcal{M}({\mathcal{N}}_{G_1}(x),\dots ,{\mathcal{N}}_{G_l}(x))(x\in I^n).$$ Our results provide a unified approach to the celebrated inequalities obtained by Kedlaya in 1994 and by Leng--Si--Zhu in 2004 and offer also new families of mixed-mean inequalities.