We establish some multiple integral Hilbert-Pachpatte-type inequalities. As applications, we get some inverse forms of Pachpatte's inequalities which were established in 1998.
1. Introduction
In 1934, Hilbert [1] established the following well-known integral inequality.
If , , , and , then
where is the best value.
In recent years, considerable attention has been given to various extensions and improvements of the Hilbert inequality form different viewpoints [2–10]. In particular, Pachpatte [11] proved some inequalities similar to Hilbert's integral inequalities in 1998. In this paper, we establish some new multiple integral Hilbert-Pachpatte-type inequalities.
2. Main Results
Theorem 2.1.
Let , let , , where are positive real numbers, and define , for . Then for , and ,
Proof.
From the hypotheses and in view of inverse Hölder integral inequality (see [12]), it is easy to observe that
Let us note the following means inequality:
We obtain that
Integrating both sides of (2.4) over from to and using the special case of inverse Hölder integral inequality, we observe that
The proof is complete.
Remark 2.2.
Taking , to (2.1), (2.1) changes to
This is just an inverse inequality similar to the following inequality which was proved by Pachpatte [11]:
Theorem 2.3.
Let , , , and be as in Theorem 2.1. Let be n positive functions defined for , and define where are positive real numbers. Let be n real-valued nonnegative, concave, and super-multiplicative functions defined on . Then
where
Proof.
By using Jensen integral inequality (see [11]) and inverse Hölder integral inequality (see [12]) and noticing that are real-valued super-multiplicative functions, it is easy to observe that
In view of the means inequality and integrating two sides of (2.10) over from to and noticing Hölder integral inequality, we observe that
This completes the proof of Theorem 2.3.
Remark 2.4.
Taking , to (2.8), (2.8) changes to
where
This is just an inverse inequality similar to the following inequality which was proved by Pachpatte [11]:
where
Theorem 2.5.
Let , , , , and be as Theorem 2.3, and define for , where are positive real numbers. Let be real-valued, nonnegative, and concave functions on . Then
Proof.
From the hypotheses and by using Jensen integral inequality and the inverse Hölder integral inequality, we have
Hence
Remark 2.6.
Taking , to (2.16), (2.16) changes to
This is just an inverse inequality similar to the following inequality which was proved by Pachpatte [11]:
Remark 2.7.
In (2.20), if , then , . Therefore (2.20) changes to
This is just an inverse inequality similar to the following Inequality which was proved by Pachpatte [11]:
Acknowledgments
This paper is supported by the National Natural Sciences Foundation of China (10971205). This paper is partially supported by the Research Grants Council of the Hong Kong SAR, China (Project no. HKU7016/07P) and an HKU Seed Grant for Basic Research.
References
-
Hardy, GH, Littlewood, JE, Pólya, G: Inequalities,p. xii+324. Cambridge University Press, Cambridge, Mass, USA (1952)
-
Yang, B: On a relation between Hilbert's inequality and a Hilbert-type inequality. Applied Mathematics Letters. 21(5), 483–488 (2008). Publisher Full Text
-
Anastassiou, GA: Hilbert-Pachpatte type general multivariate integral inequalities. International Journal of Applied Mathematics. 20(4), 549–573 (2007)
-
Yang, BC: Hilbert's inequality with some parameters. Acta Mathematica Sinica. 49(5), 1121–1126 (2006)
-
Kuang, JC, Debnath, L: The general form of Hilbert's inequality and its converses. Analysis Mathematica. 31(3), 163–173 (2005). Publisher Full Text
-
Zhao, C-J, Cheung, W-S: Sharp integral inequalities involving high-order partial derivatives. Journal of Inequalities and Applications. 2008, (2008)
-
Changjian, Z, Pecarić, J, Gangsong, L: Inverses of some new inequalities similar to Hilbert's inequalities. Taiwanese Journal of Mathematics. 10(3), 699–712 (2006)
-
Yang, B: On new generalizations of Hilbert's inequality. Journal of Mathematical Analysis and Applications. 248(1), 29–40 (2000). Publisher Full Text
-
Kuang, JC: On new extensions of Hilbert's integral inequality. Journal of Mathematical Analysis and Applications. 235(2), 608–614 (1999). Publisher Full Text
-
Gao, M, Yang, B: On the extended Hilbert's inequality. Proceedings of the American Mathematical Society. 126(3), 751–759 (1998). Publisher Full Text
-
Pachpatte, BG: On some new inequalities similar to Hilbert's inequality. Journal of Mathematical Analysis and Applications. 226(1), 166–179 (1998). Publisher Full Text
-
Beckenbach, EF, Bellman, R: Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Bd. 30,p. xii+198. Springer, Berlin, Germany (1961)