Characterization of the Hardy property of means and the best Hardy constants

Abstract

The aim of this paper is to characterize in broad classes of means the so-called Hardy means, i.e., those means $M:{\bigcup }_{n=1}^{\infty }{\mathbb{R}}_{+}^n\to {\mathbb{R}}_{+}$ that satisfy the inequality $$\sum _{n=1}^{\infty }M(x_1,\dots ,x_n)\le C\sum _{n=1}^{\infty }{x}_n$$ for all positive sequences $(x_n)$ with some finite positive constant $C$. One of the main results offers a characterization of Hardy means in the class of symmetric, increasing, Jensen concave and repetition invariant means and also a formula for the best constant $C$ satisfying the above inequality.

References in corpus (2)

• Computer aided solution of the invariance equation for two-variable Gini means
• Computer aided solution of the invariance equation for two-variable Stolarsky means

Cited by in corpus (9)

• 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 quasideviation means
• On Hardy type inequalities for weighted means
• Online premeans and their computation complexity
• Weakening of Hardy property for means
• Interval-type theorems concerning means
• On the integral approach to means and their Hardy property
• On properties of weighted Hardy constant for means