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

For every symmetric mean $\mathscr{M} \colon \bigcup_{n=1}^\infty I^n \to I$ (where $I$ an interval) and a nonzero function $W \colon \{1,\dots,n\} \to \mathbb{N} \cup \{0\}$, define an $n$-variable mean by $$\mathscr{M}_W(x):=\mathscr{M}\big(\underbrace{x_1,\dots,x_1}_{W(1)\text{-times}},\dots,\underbrace{x_n,\dots,x_n}_{W(n)\text{-times}}\big) \text{ for }x=(x_1,\dots,x_n) \in I^n.$$ Given two symmetric means $\mathscr{M},\,\mathscr{N} \colon \bigcup_{n=1}^\infty I^n \to I$ satisfying the so-called Ingham--Jessen inequality and some nonzero functions $F_1,\dots,F_k$, $G_1,\dots,G_l \colon \{1,\dots,n\} \to \mathbb{N} \cup \{0\}$, we establish sufficient conditions for inequalities of the form $$\mathscr{N} \big( \mathscr{M}_{F_1}(x),\dots,\mathscr{M}_{F_k}(x)\big) \le \mathscr{M} \big( \mathscr{N}_{G_1}(x),\dots,\mathscr{N}_{G_l}(x)\big) \qquad(x \in I^n).$$ 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.