Open Access

A Hilbert's Inequality with a Best Constant Factor

Journal of Inequalities and Applications20092009:820176

Received: 6 February 2009

Accepted: 23 July 2009

Published: 23 August 2009


We give a new Hilbert's inequality with a best constant factor and some parameters.


Constant FactorWeight CoefficientEquivalent FormStrict InequalityIntegral Inequality

1. Introduction

If , , such that and , then the well-known Hardy-Hilbert's inequality and its equivalent form are given by

where the constant factors are all the best possible [1]. It attracted some attention in the recent years. Actually, inequalities (1.1) and (1.2) have many generalizations and variants. Equation (1.1) has been strengthened by Yang and others ( including integral inequalities ) [211].

In 2006, Yang gave an extension of [2] as follows.

If , , such that , , then
is the Beta function.

In 2007 Xie gave a new Hilbert-type Inequality [3] as follows.

If , and the right of the following inequalities converges to some positive numbers, then


The main objective of this paper is to build a new Hilbert's inequality with a best constant factor and some parameters.

In the following, we always suppose that

(1) , , ,

(2)both functions and are differentiable and strict increasing in and respectively,

(3) are strictly increasing in and respectively. is strict decreasing on and ,


2. Some Lemmas

Lemma 2.1.

Define the weight coefficients as follows:


Let then if then
if ), then

On the other hand, . Setting , then Similarly, .

Lemma 2.2.

For one has



The lemma is proved.

Lemma 2.3.

Setting (or and (or , resp.), then is strictly decreasing, then

There for any ).


We have

Easily, had up bounded when

3. Main Results

Theorem 3.1.

If , , , then
is defined by Lemma 2.1.


By Hölder's inequality [12] and (2.5),
setting By(3.1) we have
By and (3.4) taking the form of strict inequality, we have (3.1). By Hölder's inequality[12], we have

as . By (3.2), (3.5) taking the form of strict inequality, we have (3.1).

Theorem 3.2.

If , then both constant factors, and of (3.1) and (3.2), are the best possible.


We only prove that is the best possible. If the constant factor in (3.1) is not the best possible, then there exists a positive (with ), such that
For , setting , then
On the other hand and ),
By (3.6), (3.7), (3.8), and Lemma 2.3, we have

We have , . This contracts the fact that .

Authors’ Affiliations

Department of Mathematics, Shaoguan University, Shaoguan, China
Department of Mathematics, Zhaoqing University, Zhaoqing, China


  1. Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge, UK; 1952:xii+324.MATHGoogle Scholar
  2. Yang BC: On Hilbert's inequality with some parameters. Acta Mathematica Sinica. Chinese Series 2006,49(5):1121–1126.MathSciNetMATHGoogle Scholar
  3. Xie Z: A new Hilbert-type inequality with the kernel of --homogeneous. Journal of Jilin University. Science Edition 2007,45(3):369–373.MathSciNetMATHGoogle Scholar
  4. Xie Z, Yang B: A new Hilbert-type integral inequality with some parameters and its reverse. Kyungpook Mathematical Journal 2008,48(1):93–100.MathSciNetView ArticleMATHGoogle Scholar
  5. Yang B: A Hilbert-type inequality with a mixed kernel and extensions. Journal of Sichuan Normal University. Natural Science 2008,31(3):281–284.MATHGoogle Scholar
  6. Xie Z, Zeng Z: A Hilbert-type integral with parameters. Journal of Xiangtan University. Natural Science 2007,29(3):24–28.MATHGoogle Scholar
  7. Wenjie W, Leping H, Tieling C: On an improvenment of Hardy-Hilbert's type inequality with some parameters. Journal of Xiangtan University. Natural Science 2008,30(2):12–14.Google Scholar
  8. Xie Z: A new reverse Hilbert-type inequality with a best constant factor. Journal of Mathematical Analysis and Applications 2008,343(2):1154–1160. 10.1016/j.jmaa.2008.02.007MathSciNetView ArticleMATHGoogle Scholar
  9. Yang B: On an extended Hardy-Hilbert's inequality and some reversed form. International Mathematical Forum 2006,1(37–40):1905–1912.MathSciNetMATHGoogle Scholar
  10. Xie Z: A Hilbert-type inequality with the kernel of irrational expression. Mathematics in Practice and Theory 2008,38(16):128–133.MathSciNetGoogle Scholar
  11. Xie Z, Rong JM: A new Hilbert-type inequality with some parameters. Journal of South China Normal University. Natural Science Edition 2008,120(2):38–42.MathSciNetMATHGoogle Scholar
  12. Kang J: Applied Inequalities. Shangdong Science and Technology Press, Jinan, China; 2004.Google Scholar


© Z. Zeng and Z.-t. Xie. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.