Abstract
In the present paper we establish new inequalities similar to the extensions of Hilbert’s double-series inequality and also give their integral analogues. Our results provide some new estimates to these types of inequalities.
MSC:26D15.
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1 Introduction
In recent years several authors have given considerable attention to Hilbert’s double-series inequality together with its integral version, inverse version, and various generalizations (see [1–9]). In this paper, we establish multivariable sum inequalities for the extensions of Hilbert’s inequality and also obtain their integral forms. Our results provide some new estimates to these types of inequalities.
The well-known classical extension of Hilbert’s double-series theorem can be stated as follows [10], p.253].
Theorem A Ifare real numbers such thatand, where, as usual, andare the conjugate exponents ofandrespectively, then
wheredepends onandonly.
In 2000, Pachpatte [11] established a new inequality similar to inequality (1.1) as follows:
Theorem A′ Let p qandbe as in [11], then
The integral analogue of inequality (1.1) is as follows [10], p.254].
Theorem B Let p, q, , and λ be as in Theorem A. Ifand, then
wheredepends on p and q only.
In [11], Pachpatte also established a similar version of inequality (1.3) as follows.
Theorem B′ Let p qandbe as in [11], then
In the present paper we establish some new inequalities similar to Theorems A, A′, B and B′. Our results provide some new estimates to these types of inequalities.
2 Statement of results
Our main results are given in the following theorems.
Theorem 2.1 Letbe constants and. Letbe real-valued functions defined for, where () are natural numbers. For convenience, we writeand. Define the operatorsbyfor any function. Then
where
Remark 2.1 Let change to in Theorem 2.1 and in view of and for any function , , then
where
Remark 2.2 Taking for in Remark 2.1. If satisfy and , then inequality (2.2) reduces to
which is an interesting variation of inequality (1.1).
On the other hand, if , then and so , . In this case inequality (2.3) reduces to
This is just a similar version of inequality (1.2) in Theorem A′.
Theorem 2.2 Letbe constants and. Letbe real-valued nth differentiable functions defined on, where, and. Suppose
then
where
Remark 2.3 Let change to in Theorem 2.2 and in view of , , then
where
Remark 2.4 Taking for in Remark 2.3, if are such that and , inequality (2.5) reduces to
which is an interesting variation of inequality (1.3).
On the other hand, if , then and so , . In this case inequality (2.6) reduces to
This is just a similar version of inequality (1.4) in Theorem B′.
3 Proofs of results
Proof of Theorem 2.1 From the hypotheses , we have
From the hypotheses of Theorem 2.1 and in view of Hölder’s inequality (see [10]) and inequality for mean [10], we obtain
Dividing both sides of (3.2) by and then taking sums over from 1 to (), respectively and then using again Hölder’s inequality, we obtain
This concludes the proof. □
Proof of Theorem 2.2 From the hypotheses of Theorem 2.2, we have
On the other hand, by using Hölder’s integral inequality (see [10]) and the following inequality for mean [10],
we obtain
Dividing both sides of (3.4) by and then integrating the result inequality over from 1 to (), respectively and then using again Hölder’s integral inequality, we obtain
This concludes the proof. □
References
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Acknowledgement
CJZ is supported by National Natural Science Foundation of China (10971205). WSC is partially supported by a HKU URG grant. The authors express their grateful thanks to the referees for their many very valuable suggestions and comments.
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The authors declare that they have no competing interests.
Authors’ contributions
C-JZ and W-SC jointly contributed to the main results Theorems 2.1 and 2.2. All authors read and approved the final manuscript.
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Zhao, CJ., Cheung, WS. On Hilbert type inequalities. J Inequal Appl 2012, 145 (2012). https://doi.org/10.1186/1029-242X-2012-145
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DOI: https://doi.org/10.1186/1029-242X-2012-145