On Cèsaro and Copson norms of nonnegative sequences

Abstract

The C`esaro and Copson norms of a nonnegative sequence are lp-norms of its arithmetic means and the corresponding conjugate means. It is well known that, for $1 < p < \infty$, these norms are equivalent. In 1996, G. Bennett posed the problem of finding the best constants in the associated inequalities. The solution of this problem requires the evaluation of four constants. Two of them were found by G. Bennett. We find one of the two unknown constants and also prove one optimal weighted-type estimate regarding the remaining constant.