On Hilbert type inequalities

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.

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 AIfare 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′Letpqandbe as in[11], then

The integral analogue of inequality (1.1) is as follows [10], p.254].

Theorem BLetp, q, , andλbe as in Theorem A. Ifand, then

wheredepends onpandqonly.

In [11], Pachpatte also established a similar version of inequality (1.3) as follows.

Theorem B′Letpqandbe 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.

Our main results are given in the following theorems.

Theorem 2.1Letbe constants and. Letbe real-valued functions defined for, where () are natural numbers. For convenience, we writeand. Define the operatorsbyfor any function. Then

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 2.2Letbe constants and. Letbe real-valuednth differentiable functions defined on, where, and. Suppose

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^′.

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. □

The authors declare that they have no competing interests.

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.

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.

1. Pachpatte, BG: On some new inequalities similar to Hilbert’s inequality. J. Math. Anal. Appl.. 226, 166–179 (1998). Publisher Full Text

2. Handley, GD, Koliha, JJ, Pečarić, JE: New Hilbert-Pachpatte type integral inequalities. J. Math. Anal. Appl.. 257, 238–250 (2001). Publisher Full Text

3. Gao, MZ, Yang, BC: On the extended Hilbert’s inequality. Proc. Am. Math. Soc.. 126, 751–759 (1998). Publisher Full Text

4. Kuang, JC: On new extensions of Hilbert’s integral inequality. J. Math. Anal. Appl.. 235, 608–614 (1999). Publisher Full Text

5. Yang, BC: On new generalizations of Hilbert’s inequality. J. Math. Anal. Appl.. 248, 29–40 (2000). Publisher Full Text

6. Zhao, CJ: On inverses of disperse and continuous Pachpatte’s inequalities. Acta Math. Sin.. 46, 1111–1116 (2003)

7. Zhao, CJ: Generalizations on two new Hilbert type inequalities. J. Math. (Wuhan). 20, 413–416 (2000)

8. Zhao, CJ, Debnath, L: Some new inverse type Hilbert integral inequalities. J. Math. Anal. Appl.. 262, 411–418 (2001). Publisher Full Text

9. Handley, GD, Koliha, JJ, Pečarić, JE: A Hilbert type inequality. Tamkang J. Math.. 31, 311–315 (2000)

10. Hardy, GH, Littlewood, JE, Pólya, G: Inequalities, Cambridge University Press, Cambridge (1934)

11. Pachpatte, BG: Inequalities similar to certain extensions of Hilbert’s inequality. J. Math. Anal. Appl.. 243, 217–227 (2000). Publisher Full Text