LAUSR

AbstractIn 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean M$\mathscr{M}$ the Kedlaya-type inequality A(x1,M(x1,x2),…,M(x1,…,xn))≤M(x1,A(x1,x2),…,A(x1,…,xn))$$ \mathscr{A} \bigl(x_{1},\mathscr{M}(x_{1},x_{2}), \ldots,\mathscr{M}(x _{1},\ldots,x_{n}) \bigr) \le \mathscr{M} \bigl( x_{1}, \mathscr{A}(x _{1},x_{2}), \ldots,\mathscr{A}(x_{1},\ldots,x_{n}) \bigr) $$ holds for an arbitrary (xn)$(x_{n})$ (A$\mathscr{A}$ stands for the arithmetic mean). We are going to prove the weighted counterpart of this inequality. More precisely, if (xn)$(x_{n})$ is a vector with corresponding (non-normalized) weights (λn)$(\lambda_{n})$ and Mi=1n(xi,λi)$\mathscr{M}_{i=1}^{n}(x _{i},\lambda_{i})$ denotes the weighted mean then, under analogous conditions on M$\mathscr{M}$, the inequality holds for every (xn)$(x_{n})$ and (λn)$(\lambda_{n})$ such that the sequence (λkλ1+⋯+λk)$(\frac{\lambda_{k}}{\lambda_{1}+\cdots+\lambda_{k}})$ is decreasing.