On Hardy-type inequalities for weighted means

Abstract

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 $({\lambda }_n{)}_{n=1}^{\infty }$ and a weighted mean $\mathcal{M}$, we search for the smallest extended real number $C$ such that

$${\sum }_{n=1}^{\infty }{\lambda }_n\mathcal{M}((x_1,\dots ,x_n),({\lambda }_1,\dots ,{\lambda }_n))\le C{\sum }_{n=1}^{\infty }{\lambda }_n{x}_n\text{for all}x\in {\ell }_1(\lambda ).$$ The main results provide a complete answer in the case when $\mathcal{M}$ is monotone and satisfies the weighted counterpart of the Kedlaya inequality. In particular, this is the case if $\mathcal{M}$ is symmetric, concave, and the sequence $(\frac{{\lambda }_n}{{\lambda }_1+\cdots +{\lambda }_n}{)}_{n=1}^{\infty }$ is nonincreasing. In addition, we prove that if $\mathcal{M}$ is a symmetric and monotone mean, then the biggest possible weighted Hardy constant is achieved if $\lambda$ is the constant vector.