Hardy-type inequalities via auxiliary sequences

Abstract

We prove some Hardy-type inequalities via an approach that involves constructing auxiliary sequences.

On l p norms of weighted mean matrices

“…Recently, the author has given a simple proof [16] of the result of Levin and Stečkin and also extended their result to a case where p is slightly bigger than 1/3. In general, integral Hardy-type inequalities suggest that various inequalities in the following forms or their reverses should hold with λ n = n α and Λ n defined as in (1.3) for different choices of p and α with U some constants depending on p and λ n 's whose values are suggested by the integral cases:…”

“…The p > 1 case has been extensively studied in [14], [8], [16] and will also be our main focus in the paper. We now take a look at the inequalities (2.7) for 0 < p < 1.…”

“…Integral inequalities (2.4) suggest that the reversed inequalities of (2.7) hold for the cases 0 < p < 1, 0 < 1 + α < 1/p (note that we get back (2.5) for the special case α = 0 of λ n = n α ). We now treat these cases for general λ k 's following the method in Section 3 of [16]. It is then easy to see that inequalities (2.7) hold for the cases 0 < p < 1, 0…”

“…Among the many different proofs of Hardy's inequality (1.1) as well as its generalizations and extensions in the literature, there are notably Kaluza and Szegö's approach [20] (see also [21]) and Redheffer's "recurrent inequalities" [27]. It is shown in [16] that these two methods above are essentially the same (in [16], we credited the approach of Kaluza and Szegö to Knopp but apparently the paper [20] is earlier) and we shall further show in this paper that Kaluza and Szegö's approach can be regarded as an approximation to Carleman's approach in Section 3. Hence instead of Carleman's approach, there is not much lost using Kaluza and Szegö's or Redheffer's approach when studying Hardy-type inequalities, yet technically they are much easier to handle.…”

“…The proofs of (1.10) in both [14] and [8] are the same, they both use the result of Cartlidge (Theorem 1.1 above). Recently, the author [16] has shown that inequalities (1.10) hold for p ≥ 2, 1 ≤ α ≤ 1 + 1/p or 1 < p ≤ 4/3, 1 + 1/p ≤ α ≤ 2.…”