On properties of weighted Hardy constant for means

Abstract

For a given weighted mean $\mathcal{M}$ defined on a subinterval of ${\mathbb{R}}_{+}$ and a sequence of weights $\lambda =({\lambda }_n{)}_{n=1}^{\infty }$ we define a Hardy constant $\mathcal{H}(\lambda )$ as the smallest extended real number such that $$\sum _{n=1}^{\infty }{\lambda }_n\mathcal{M}((x_1,\dots ,x_n),({\lambda }_1,\dots ,{\lambda }_n))\le \mathcal{H}(\lambda )\cdot \sum _{n=1}^{\infty }{\lambda }_n{x}_n\text{ for all }x\in {\ell }^1(\lambda ).$$ The aim of this note is to present a comprehensive study of the mapping $\mathcal{H}$. For example we prove that it is lower semicontinuous in the pointwise topology. Moreover we show that whenever $\mathcal{M}$ is a monotone and Jensen-concave mean which is continuous in its weights then $\mathcal{H}$ is monotone with respect to the partitioning of the vector. Finally we deliver some sufficient conditions for $\lambda$ to validate the equality $\mathcal{H}(\lambda )=\sup \mathcal{H}$ for every symmetric and monotone mean.

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References in corpus (6)

• Characterization of the Hardy property of means and the best Hardy constants
• On the best Hardy constant for quasi-arithmetic means and homogeneous deviation means
• On Kedlaya type inequalities for weighted means
• On Hardy type inequalities for weighted means
• On Hardy type inequalities for weighted quasideviation means
• Weakening of Hardy property for means