Our aim in this article is to establish weighted Hardy-type inequalities in a broad family of means. In other words, for a fixed vector of weights and a weighted mean , we search for the smallest extended real number such that
The main results provide a complete answer in the case when is monotone and satisfies the weighted counterpart of the Kedlaya inequality. In particular, this is the case if is symmetric, concave, and the sequence is nonincreasing. In addition, we prove that if is a symmetric and monotone mean, then the biggest possible weighted Hardy constant is achieved if is the constant vector.
"On Hardy-type inequalities for weighted means." Banach J. Math. Anal. 13 (1) 217 - 233, January 2019. https://doi.org/10.1215/17358787-2018-0023