On properties of weighted Hardy constant for means

For a given weighted mean $\mathscr{M}$ defined on a subinterval of $\mathbb{R}_+$ and a sequence of weights $\lambda=(\lambda_n)_{n=1}^\infty$ we define a Hardy constant $\mathscr H(\lambda)$ as the smallest extended real number such that $$\sum_{n=1}^\infty \lambda_n \mathscr{M}\big((x_1,\dots,x_n),(\lambda_1,\dots,\lambda_n)\big) \le \mathscr H(\lambda) \cdot \sum_{n=1}^\infty \lambda_n x_n \text{ for all }x \in \ell^1(\lambda).$$ The aim of this note is to present a comprehensive study of the mapping $\mathscr H$. For example we prove that it is lower semicontinuous in the pointwise topology. Moreover we show that whenever $\mathscr{M}$ is a monotone and Jensen-concave mean which is continuous in its weights then $\mathscr H$ is monotone with respect to the partitioning of the vector. Finally we deliver some sufficient conditions for $\lambda$ to validate the equality $\mathscr H(\lambda)=\sup \mathscr H$ for every symmetric and monotone mean.