On Carleman and Knopp's Inequalities

Abstract

A sharpened version of Carleman's 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 Carleman's inequality, which is usually referred to as Knopp's inequality. A new elementary proof of (Carleman-)Knopp's inequality and a new inequality of Hardy-Knopp type is pointed out. # 2002 Elsevier Science (USA)

Hardy-Type Inequalities on Time Scale via Convexity in Several Variables

“…More general Hardy integral inequalities have been studied in continuous cases. We notice only [2][3][4][5][6] and the references therein.…”

Some new refinements of strengthened Hardy and Pólya–Knopp′s inequalities

“…This result, the so-called weighted Hardy-Knopp-type inequality, was already obtained in [9, Theorem 1], while its particular case (1.5), originally proved in [18], follows by setting u(x) ≡ 1 . Therefore, (2.14) may be regarded as a refined weighted inequality of the Hardy-Knopp type and relation (2.3) as its generalization.…”

“…in the monographs [13,22,23,25,26,27,28], expository papers [6,17,21], and the references cited therein. Besides, here we also emphasize the papers [2,4,5,7,8,9,18,19,24,29,32,33], all of which to some extent have guided us in the research we present here.…”

“…In particular, in 1970, R. P. Boas [2] proved that (1.1) and (1.3) are just special cases of a much more general inequality On the other hand, obviously unaware of the mentioned more general Boas's result for Hardy-Littlewood averages, in 2002, S. Kaijser et al [18] established the so-called general Hardy-Knopp-type inequality for positive functions f : R + −→ R,…”

A Strengthened Form of a General Hardy-Type Inequality Obtained via Superquadraticity and Its Applications

“…Inequality (24) was extensively studied in paper [12], but was known even before (see e.g. Godunova [7]).…”