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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 925464, 4 pages
http://dx.doi.org/10.1155/2013/925464
Research Article

A Generalization on Some New Types of Hardy-Hilbert’s Integral Inequalities

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathum Thani 12121, Thailand

Received 15 May 2013; Accepted 17 September 2013

Academic Editor: Wilfredo Urbina

Copyright © 2013 Banyat Sroysang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Sulaiman presented, in 2008, new kinds of Hardy-Hilbert’s integral inequality in which the weight function is homogeneous. In this paper, we present a generalization on the kinds of Hardy-Hilbert’s integral inequality.

1. Introduction and Preliminaries

For any two nonnegative measurable functions and such that we have the Hilbert’s integral inequality [1] that The constant is the best possible. In 1925, Hardy [2] extended the Hilbert’s integral inequality into the integral inequality as follows. If , , and such that then we have the Hardy-Hilbert’s integral inequality that The constant is the best possible. Both the two inequalities are important in mathematical analysis and its applications [3].

In 1938, Widder [4] studied on the Stieltjes Transform .

Now, we recall the beta function as follows:

In 2001, Yang [5] extended the Hardy-Hilbert’s integral inequality into the following integral inequality. If , , , and such that

then we have where . The constant is the best possible.

We also recall that a nonnegative function which is said to be homogeneous function of degree if for all . And we say that is increasing if and are increasing functions.

In 2008, Sulaiman [6] gave new integral inequality similar to the Hardy-Hilbert’s integral inequality. If , , , , is a positive increasing homogeneous function of degree , and and then, for all , we have where

In this paper, we present a generalization of the integral inequality (9) and its applications. Next proposition will be used in the next section.

Proposition 1 (see [6]). Let be a positive increasing function, and . Then, for all , one has

2. Main Results

Theorem 2. Let , , , , and let be positive increasing homogeneous function of degree , and and
and let be a function such that for all .
Then, for all , one has where

Proof. Let and .
By the Hölder inequality, the assumption of , and the Tonelli theorem, we have
Now, we put and for the first integral, and then we put and for the second integral.
And, by Proposition 1, one has
Then, by the assumption, one has
This proof is completed.

3. Applications

Corollary 3. Let , , and , and let be a positive increasing homogeneous function of degree , and and
Then, for all , one has where

Proof. (a) This follows from Theorem 2 where for all .
(b) This follows from Theorem 2 where for all .
(c) This follows from Theorem 2 where for all .
(d) This follows from Theorem 2 where for all .

4. Open Problem

In this section, we pose a question that is how to generalize the integral inequality (13) if may not satisfy the property for all .

Acknowledgments

The author would like to thank the referees for their useful comments and suggestions.

References

  1. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 2nd edition, 1952. View at MathSciNet
  2. G. H. Hardy, “Notes on a theorem of Hilbert concerning series of positive terms,” Proceedings of the London Mathematical Society, vol. 23, pp. 45–46, 1925. View at Google Scholar
  3. D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, vol. 53 of Mathematics and Its Applications (East European Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. View at MathSciNet
  4. D. V. Widder, “The Stieltjes transform,” Transactions of the American Mathematical Society, vol. 43, no. 1, pp. 7–60, 1938. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. B. Yang, “On Hardy-Hilbert’s integral inequality,” Journal of Mathematical Analysis and Applications, vol. 261, pp. 295–306, 2001. View at Google Scholar
  6. W. T. Sulaiman, “A study on some new types of Hardy-Hilbert's integral inequalities,” Banach Journal of Mathematical Analysis, vol. 2, no. 1, pp. 16–20, 2008. View at Google Scholar · View at MathSciNet