Open Access

On Inverse Hilbert-Type Inequalities

Journal of Inequalities and Applications20072008:693248

DOI: 10.1155/2008/693248

Received: 14 November 2007

Accepted: 4 December 2007

Published: 10 December 2007

Abstract

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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq1_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq2_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq3_HTML.gif be https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq4_HTML.gif positive sequences of real numbers defined for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq5_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq6_HTML.gif are natural numbers, define https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq7_HTML.gif , and define https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq8_HTML.gif . Then for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq10_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq11_HTML.gif , one has

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ1_HTML.gif
(2.1)

Proof.

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

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ2_HTML.gif
(2.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq13_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq14_HTML.gif , we obtain that

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ3_HTML.gif
(2.3)

thus

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ4_HTML.gif
(2.4)

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

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ5_HTML.gif
(2.5)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ6_HTML.gif
(2.6)

Taking the sum of both sides of (2.6) over https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq15_HTML.gif from 1 to https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq16_HTML.gif first and then using again inverse Hölder's inequality, we obtain that

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ7_HTML.gif
(2.7)

This completes the proof.

Remark 2.2.

Taking https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq17_HTML.gif to (2.1), (2.1) becomes

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ8_HTML.gif
(2.8)

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

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ9_HTML.gif
(2.9)

Theorem 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq18_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq19_HTML.gif be as defined in Theorem 2.1. Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq20_HTML.gif be https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq21_HTML.gif positive sequences for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq22_HTML.gif Set https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq23_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq24_HTML.gif be https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq25_HTML.gif real-valued nonnegative, concave, and supermultiplicative functions defined on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq26_HTML.gif Then,

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ10_HTML.gif
(2.10)

where

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ11_HTML.gif
(2.11)

Proof.

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

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ12_HTML.gif
(2.12)

Dividing both sides of (2.12) by https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq27_HTML.gif and then taking the sum over https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq28_HTML.gif from 1 to https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq29_HTML.gif (and in view of inverse Hölder's inequality), we have

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ13_HTML.gif
(2.13)

The proof is complete.

Remark 2.4.

Taking https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq30_HTML.gif to (2.10), (2.10) becomes

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ14_HTML.gif
(2.14)

where

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ15_HTML.gif
(2.15)

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

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ16_HTML.gif
(2.16)

where

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ17_HTML.gif
(2.17)

Similarly, the following theorem also can be established.

Theorem 2.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq31_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq32_HTML.gif be as in Theorem 2.3 and define https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq33_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq34_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq35_HTML.gif be https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq36_HTML.gif real-valued, nonnegative, and concave functions defined on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq37_HTML.gif Then,

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ18_HTML.gif
(2.18)

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 https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq38_HTML.gif to (2.18), (2.18) becomes

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ19_HTML.gif
(2.19)

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

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ20_HTML.gif
(2.20)

Remark 2.7.

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

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ21_HTML.gif
(2.21)

Accordingly, in the special case when https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq39_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq40_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq41_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_IEq42_HTML.gif , then the inequality (2.18) reduces to the following inequality:

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ22_HTML.gif
(2.22)

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

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ23_HTML.gif
(2.23)

Declarations

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.

Authors’ Affiliations

(1)
Department of Information and Mathematics Sciences, College of Science, China Jiliang University
(2)
Department of Mathematics, The University of Hong Kong

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Copyright

© Z. Changjian and W.-S. Cheung. 2008

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.