This paper deals with new inversetype 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 Hilberttype 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 inversetype inequalities of the above integral inequalities. This paper deals with some new inversetype 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
Proof.
By using the following inequality (see [10, page 39]):
where , , and , we obtain that
thus
From inequality (2.4) and in view of the following mean inequality and inverse Hölder's inequality [10, page 24], we have
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
This completes the proof.
Remark 2.2.
Taking to (2.1), (2.1) becomes
This is just an inverse form of the following inequality which was proven by Pachpatte [9]:
Theorem 2.3.
Let , and be as defined in Theorem 2.1. Let be positive sequences for Set . Let be realvalued nonnegative, concave, and supermultiplicative functions defined on Then,
where
Proof.
From the hypotheses and by Jensen's inequality, the means inequality, and inverse Hölder's inequality, we obtain that
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
The proof is complete.
Remark 2.4.
Taking to (2.10), (2.10) becomes
where
This is just an inverse of the following inequality which was proven by Pachpatte [9]:
where
Similarly, the following theorem also can be established.
Theorem 2.5.
Let , and be as in Theorem 2.3 and define for . Let be realvalued, nonnegative, and concave functions defined on Then,
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
This is just an inverse of the following inequality which was proven by Pachpatte [9]:
Remark 2.7.
In view of L'Hôpital law, we have the following fact:
Accordingly, in the special case when , , and , let , then the inequality (2.18) reduces to the following inequality:
This is just a discrete form of the following inequality which was proven by Zhao and Debnath [11]:
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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