On Kedlaya-type inequalities for weighted means

In 2016 we proved that for every symmetric, repetition invariant and Jensen concave mean [Formula: see text] the Kedlaya-type inequality [Formula: see text] holds for an arbitrary [Formula: see text] ([Formula: see text] stands for the arithmetic mean). We are going to prove the weighted counterpart of this inequality. More precisely, if [Formula: see text] is a vector with corresponding (non-normalized) weights [Formula: see text] and [Formula: see text] denotes the weighted mean then, under analogous conditions on [Formula: see text] , the inequality [Image: see text] holds for every [Formula: see text] and [Formula: see text] such that the sequence [Formula: see text] is decreasing.