On Carleman and Knoppʹs Inequalities Original Research Article

A sharpened version of Carlemanʹs inequality is proved. This result unifies and generalizes some recent results of this type. Also the “ordinary” sum that serves as the upper bound is replaced by the corresponding Cesaro sum. Moreover, a Carleman-type inequality with a more general measure is proved and this result may also be seen as a generalization of a continuous variant of Carlemanʹs inequality, which is usually referred to as Knoppʹs inequality. A new elementary proof of (Carleman–)Knoppʹs inequality and a new inequality of Hardy–Knopp type is pointed out.