The connection between Hilbert and Hardy inequalities

Abstract

In this paper we introduce some new forms of the Hilbert integral inequality, and we study the connection between the obtained inequalities with Hardy inequalities. The reverse form and some applications are also given. MSC: 26D15

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]).…”

“…, namely, (15) follows. Hence, inequality (8) is equivalent to (15) and then inequalities (8), (14) and (15) are equivalent. If the constant factor in (8) is the best possible, then so is the constant factor in (14) and (15).…”

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

“…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.…”

Equivalent properties of a Mulholland-type inequality with a best possible constant factor and parameters

“…[2,3]). Recently, some new extensions with parameters and applications were given in [4][5][6][7][8][9][10][11][12][13].…”