On Kedlaya type inequalities for weighted means

In 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean $\mathcal{M}$ the Kedlaya-type inequality $$\mathcal{A}(x_1,\mathcal{M}(x_1,x_2),\dots ,\mathcal{M}(x_1,\dots ,x_n))\le \mathcal{M}(x_1,\mathcal{A}(x_1,x_2),\dots ,\mathcal{A}(x_1,\dots ,x_n))$$ holds for an arbitrary $(x_n)$ ($\mathcal{A}$ 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 ${\mathcal{M}}_{i=1}^n(x_i,{\lambda }_i)$ denotes the weighted mean then, under analogous conditions on $\mathcal{M}$, the inequality $${\mathcal{A}}_{i=1}^n({\mathcal{M}}_{j=1}^i(x_j,{\lambda }_j),{\lambda }_i)\le {\mathcal{M}}_{i=1}^n({\mathcal{A}}_{j=1}^i(x_j,{\lambda }_j),{\lambda }_i)$$ holds for every $(x_n)$ and $({\lambda }_n)$ such that the sequence $(\frac{{\lambda }_k}{{\lambda }_1+\cdots +{\lambda }_k})$ is decreasing.