Open Access Research Article

On Inverse Hilbert-Type Inequalities

Zhao Changjian1* and Wing-Sum Cheung2

Author Affiliations

1 Department of Information and Mathematics Sciences, College of Science, China Jiliang University, Hangzhou 310018, China

2 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong

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Journal of Inequalities and Applications 2008, 2008:693248  doi:10.1155/2008/693248


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2008/1/693248


Received: 14 November 2007
Revisions received: 1 December 2007
Accepted: 4 December 2007
Published: 10 December 2007

© 2008 The Author(s).

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.

This paper deals with new inverse-type Hilbert inequalities. Our results in special cases yield some of the recent results and provide some new estimates on such types of inequalities.

1. Introduction

Considerable attention has been given to Hilbert inequalities and Hilbert-type inequalities and their various generalizations by several authors including Handley et al. [1], Minzhe and Bicheng [2], Minzhe [3], Hu [4], Jichang [5], Bicheng [6], and Zhao [7, 8]. In 1998, Pachpatte [9] gave some new integral inequalities similar to Hilbert inequality (see [10, page 226]). In 2000, Zhao and Debnath [11] established some inverse-type inequalities of the above integral inequalities. This paper deals with some new inverse-type Hilbert inequalities which provide some new estimates on such types of inequalities.

2. Main Results

Theorem 2.1.

Let and . Let be positive sequences of real numbers defined for , where are natural numbers, define , and define . Then for , or , one has

(21)

Proof.

By using the following inequality (see [10, page 39]):

(22)

where , , and , we obtain that

(23)

thus

(24)

From inequality (2.4) and in view of the following mean inequality and inverse Hölder's inequality [10, page 24], we have

(25)

(26)

Taking the sum of both sides of (2.6) over from 1 to first and then using again inverse Hölder's inequality, we obtain that

(27)

This completes the proof.

Remark 2.2.

Taking to (2.1), (2.1) becomes

(28)

This is just an inverse form of the following inequality which was proven by Pachpatte [9]:

(29)

Theorem 2.3.

Let , and be as defined in Theorem 2.1. Let be positive sequences for Set . Let be real-valued nonnegative, concave, and supermultiplicative functions defined on Then,

(210)

where

(211)

Proof.

From the hypotheses and by Jensen's inequality, the means inequality, and inverse Hölder's inequality, we obtain that

(212)

Dividing both sides of (2.12) by and then taking the sum over from 1 to (and in view of inverse Hölder's inequality), we have

(213)

The proof is complete.

Remark 2.4.

Taking to (2.10), (2.10) becomes

(214)

where

(215)

This is just an inverse of the following inequality which was proven by Pachpatte [9]:

(216)

where

(217)

Similarly, the following theorem also can be established.

Theorem 2.5.

Let , and be as in Theorem 2.3 and define for . Let be real-valued, nonnegative, and concave functions defined on Then,

(218)

The proof of Theorem 2.5 can be completed by following the same steps as in the proof of Theorem 2.3 with suitable changes. Here, we omit the details.

Remark 2.6.

Taking to (2.18), (2.18) becomes

(219)

This is just an inverse of the following inequality which was proven by Pachpatte [9]:

(220)

Remark 2.7.

In view of L'Hôpital law, we have the following fact:

(221)

Accordingly, in the special case when , , and , let , then the inequality (2.18) reduces to the following inequality:

(222)

This is just a discrete form of the following inequality which was proven by Zhao and Debnath [11]:

(223)

Acknowledgments

The authors cordially thank the anonymous referee for his/her valuable comments which lead to the improvement of this paper. Research is supported by Zhejiang Provincial Natural Science Foundation of China, Grant no. Y605065, Foundation of the Education Department of Zhejiang Province of China, Grant no. 20050392, partially supported by the Research Grants Council of the Hong Kong SAR, China, Project no. HKU7016/07P.

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