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.
Categories: math.CA