Extension of Hilbert's inequality

Abstract

In this paper we make some further extensions of discrete Hilbert's inequality by using Euler-Maclaurin summation formula. We give the improvements of some previously obtained results and also compare our results with some previously known from the literature.

A generalization of Hilbert's inequality

“…In this section, we first improve the result of [7,Lemma 2] by showing that inequality (1.4) is valid for any s > 2 when λ = 1. We then deduce that lim n→∞ f s (n) = 0.…”

“…We note that the above inequality is valid when λ < 0 by the integral test. In [7,Lemma 2], it is shown that the above inequality is valid when 0 < s ≤ 2, −1 < λ < s−1 and 2 < s ≤ 14, −1 < λ ≤ 1. In the next section, we extend this result to the case of 1 < λ ≤ 2, λ + 1 < s ≤ 5 to prove the following…”

Hilbert's double series theorem extended to the case of non-homogeneous kernels

“…When K(m, n) = (m κ + n κ ) −λ , the kernel function is homogeneous of order −λ · κ. Specifying κ = λ = 1 we get Hilbert's expression in (1), while κ = 1 leads us to Yang's result (2). Assuming the homogeneity of K(·,·) and using non-conjugate Hölder pairs (p, q), p > 1, p −1 + q −1 1 we arrive at the classical Hilbert's double series theorems by Levin [5], Bonsall [1] and the recent ones by Krnić and Pečarić [4]. (We point out that the best constant problem is still open when p, q are not conjugated.…”

“…The range of the parameter λ ∈ (2 − min{p, q}, 2] in Yang's result (2) was enlarged (by new consequent parametrizations) to λ ∈ (0, 14] by Krnić and Pečarić, see [4,Theorems 1,2] in the conjugate, and [4, Theorem 4] for non-conjugate Hölder couple p, q. Therefore, the range of r in Theorem 1 can be enlarged as well.…”

“…The lonely Hilbert's double series theorems in the case of nonhomogeneous kernel remain the Mulholland's inequality [6], in which K(m, n) = ln(mn), m, n 2, and its extension by Yang, who takes Chapters 3,4]. So, the inequality result by He et al…”