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On Inverse Hilbert-Type Inequalities
Journal of Inequalities and Applications volume 2008, Article number: 693248 (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 and
. Let
be
positive sequences of real numbers defined for
, where
are natural numbers, define
, and define
. Then for
,
or
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ1_HTML.gif)
Proof.
By using the following inequality (see [10, page 39]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ2_HTML.gif)
where ,
, and
, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ3_HTML.gif)
thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ4_HTML.gif)
From inequality (2.4) and in view of the following mean inequality and inverse Hölder's inequality [10, page 24], we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ5_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ7_HTML.gif)
This completes the proof.
Remark 2.2.
Taking to (2.1), (2.1) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ8_HTML.gif)
This is just an inverse form of the following inequality which was proven by Pachpatte [9]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ9_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ10_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ11_HTML.gif)
Proof.
From the hypotheses and by Jensen's inequality, the means inequality, and inverse Hölder's inequality, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ13_HTML.gif)
The proof is complete.
Remark 2.4.
Taking to (2.10), (2.10) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ14_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ15_HTML.gif)
This is just an inverse of the following inequality which was proven by Pachpatte [9]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ16_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ17_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ19_HTML.gif)
This is just an inverse of the following inequality which was proven by Pachpatte [9]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ20_HTML.gif)
Remark 2.7.
In view of L'Hôpital law, we have the following fact:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ21_HTML.gif)
Accordingly, in the special case when ,
, and
, let
, then the inequality (2.18) reduces to the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ22_HTML.gif)
This is just a discrete form of the following inequality which was proven by Zhao and Debnath [11]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F693248/MediaObjects/13660_2007_Article_1849_Equ23_HTML.gif)
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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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Changjian, Z., Cheung, WS. On Inverse Hilbert-Type Inequalities. J Inequal Appl 2008, 693248 (2007). https://doi.org/10.1155/2008/693248
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DOI: https://doi.org/10.1155/2008/693248