Inequalities related to Hardy's and Heinig's

Abstract

In a 1975 paper [8], Heinig established the following three inequalities:where A = p/(p + s − λ) with p, s, λ real numbers satisfying p + s > λ,p > 0;where B = p/(2p + sp − λ −1) with p, s, λ real numbers satisfying 2p +sp > λ, + 1, p > 0;where is a sequence of nonnegative real numbers,and C = p[l + l/(p + s−λ)] with p, s, λ real numbers satisfying s > 0, p ≥ 1, and p +s > λ 0.

On strengthened Hardy and Pólya–Knopp's inequalities

“…Here, we just emphasize monographs [9,13,15,16], related to this topic, and mention Refs. [2][3][4][5][6]10,14,18,19], all of which to some extent have guided us in the research we present here.…”

“…First, we generalize relation (3) by adding weight functions and truncating the range of integration to ð0; bÞ: In fact, we obtain a strengthened inequality of the HardyKnopp type and also prove the so-called dual inequality to this relation, that is, an inequality with the outer integrals taken over ðb; NÞ and with the inner integral on the left-hand side taken over ðx; NÞ: Thus we unify the strengthened Hardy and Po´lya-Knopp's inequalities and derive them as special cases of our more general results. Finally, we discuss the Po´lya-Knopp's inequality, compare it to the LevinCochran-Lee's inequalities from [6,14] (cf. also [3,4,15]), and point out that these results are mutually equivalent.…”

On Carleman and Knopp's Inequalities

“…Here, we just mention the following, all of which to some extent have guided us in our investigation: Hardy [8,9] G. P ! o olya (see [4, p. 156]); Knopp [15] (see also [4, p. 487]); Carleson [6]; Redheffer [22]; Cochran and Lee [7]; Heinig [11]; Henrici [12]; Love [16]; Bicheng and Debnath [3]; Alzer [1]; Bennett [2]; Ping and Guozheng [21] and Pecaric and Stolarsky [18]. Let us just mention that some applications to continued fractions are given in [12] and that further references and information can be found in the recent interesting review article [18].…”

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

“…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.…”