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

The aim of this paper is to characterize in broad classes of means the so-called Hardy means, i.e., those means $M\colon\bigcup_{n=1}^\infty $\mathbb{R}$+^n\to$\mathbb{R}$+$thatsatisfytheinequality$${\sum }_{n=1}^{\mathrm{\infty }}M(x_1,\dots ,x_n)\le C{\sum }_{n=1}^{\mathrm{\infty }}{x}_n$$ 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.