Hilbert-type inequalities involving differential operators, the best constants, and applications

Abstract

Abstract. Motivated by some recent results, in this article we derive several Hilbert-type inequalities with a differential operator, regarding a general homogeneous kernel. Moreover, we show that the constants appearing on the right-hand sides of these inequalities are the best possible. The general results are then applied to some particular examples of homogeneous kernels and compared with previously known from the literature.Mathematics subject classification (2010): Primary 26D10, 26D15, Secondary 33B15.

On a reverse extended Hardy–Hilbert’s inequality

“…If the constant factor in (8) is the best possible, then so is the constant factor in (14) and (15).…”

“…Inequalities (1) and (3) with their extensions and reverses are important in analysis and its applications (cf. [5][6][7][8][9][10][11][12][13][14][15]).…”

On a new extended half-discrete Hilbert’s inequality involving partial sums

“…Proof As regards to the assumptions, we find 0 <λ 1 ,λ 2 < λ + 1. By (13), the unified best possible constant factor in (15) must be of the following form: By Hölder's inequality (cf. [27]), we obtain…”

“…Inequalities (1) and (2) play an important role in the analysis and its applications (cf. [2][3][4][5][6][7][8][9][10][11][12][13]). We still have the following half-discrete Hilbert-type inequality (cf.…”

“…Remark 4 (i) In (13) and (19), for 0 < λ ≤ min{p, 26}, λ 1 = λ q , λ 2 = λ p (≤ 1), we have the following equivalent inequalities:…”