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The aim of this paper is to characterize in broad classes of means theso-called Hardy means, i.e., those means $M\colon\bigcup_{n=1}^\infty\mathbb{R}_+^n\to\mathbb{R}_+$ that satisfy the inequality $$  \sum_{n=1}^\infty M(x_1,\dots,x_n) \le C\sum_{n=1}^\infty x_n $$ for allpositive sequences $(x_n)$ with some finite positive constant $C$. One of themain results offers a characterization of Hardy means in the class ofsymmetric, increasing, Jensen concave and repetition invariant means and also aformula for the best constant $C$ satisfying the above inequality.

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