Hardyʹs inequality and ultrametric matrices Original Research Article

For any p > 1 and for any sequence $\{ a_j \}?\infty_{j=1}$ of nonnegative numbers, a classical inequality of Hardy gives that$$\sum?n_{k=1} \left({\sum \nolimits?k_{i=1} a_i\over k}\right)?p\les \left({p\over p-1}\right)?p\sum?n_{k=1}a?p_k \quad {\hbox{for each}}\;n \; \in \; {\open {N}},$$unless all $a_j=0$, where the constant $[p/(p-1)]?p$ is best possible. Here, we investigate this inequality in the case p=2, and show how it can be interpreted in terms of symmetric ultrametric matrices. From this, a generalization of Hardyʹs inequality, in the case p=2, is derived.