## Banach Journal of Mathematical Analysis

### 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 $\mathscr{M}$, we search for the smallest extended real number $C$ such that

$$\sum_{n=1}^{\infty}\lambda_{n}\mathscr{M}((x_{1},\ldots ,x_{n}),(\lambda_{1},\ldots,\lambda_{n}))\le C\sum_{n=1}^{\infty}\lambda_{n}x_{n}\quad \text{for all }x\in \ell_{1}(\lambda).$$ The main results provide a complete answer in the case when $\mathscr{M}$ is monotone and satisfies the weighted counterpart of the Kedlaya inequality. In particular, this is the case if $\mathscr{M}$ 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 $\mathscr{M}$ is a symmetric and monotone mean, then the biggest possible weighted Hardy constant is achieved if $\lambda$ is the constant vector.

#### Article information

Source
Banach J. Math. Anal., Volume 13, Number 1 (2019), 217-233.

Dates
Accepted: 23 July 2018
First available in Project Euclid: 16 November 2018

https://projecteuclid.org/euclid.bjma/1542358830

Digital Object Identifier
doi:10.1215/17358787-2018-0023

Mathematical Reviews number (MathSciNet)
MR3892700

Zentralblatt MATH identifier
07002039

#### Citation

Páles, Zsolt; Pasteczka, Paweł. On Hardy-type inequalities for weighted means. Banach J. Math. Anal. 13 (2019), no. 1, 217--233. doi:10.1215/17358787-2018-0023. https://projecteuclid.org/euclid.bjma/1542358830