On Kedlaya-type inequalities for weighted means

In 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean $\mathscr{M}$ the Kedlaya-type inequality $$ \mathscr{A} \bigl(x_{1},\mathscr{M}(x_{1},x_{2}), \ldots,\mathscr{M}(x _{1},\ldots,x_{n}) \bigr) \le \mathscr{M} \bigl( x_{1}, \mathscr{A}(x _{1},x_{2}), \ldots,\mathscr{A}(x_{1},\ldots,x_{n}) \bigr) $$ holds for an arbitrary $(x_{n})$ ( $\mathscr{A}$ stands for the arithmetic mean). We are going to prove the weighted counterpart of this inequality. More precisely, if $(x_{n})$ is a vector with corresponding (non-normalized) weights $(\lambda_{n})$ and $\mathscr{M}_{i=1}^{n}(x _{i},\lambda_{i})$ denotes the weighted mean then, under analogous conditions on $\mathscr{M}$ , the inequality holds for every $(x_{n})$ and $(\lambda_{n})$ such that the sequence $(\frac{\lambda_{k}}{\lambda_{1}+\cdots+\lambda_{k}})$ is decreasing.