- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 727923, 8 pages

http://dx.doi.org/10.1155/2013/727923

## Refinements of Hardy-Type Inequalities

College of Science and Technology, North China Electric Power University, Baoding, Hebei 071051, China

Received 18 April 2013; Accepted 19 July 2013

Academic Editor: Wenchang Sun

Copyright © 2013 Jingfeng Tian and Yang-Xiu Zhou. 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

Using Hu Ke's inequality, which is a sharped Hölder's inequality, we present some new refinements of Hardy-type inequalities proposed by Imoru.

#### 1. Introduction

Let , , . Then the famous Hardy's inequality [1, Theorem 319] reads as where is nonnegative and homogeneous of degree −1. The sign of the inequality in (1) is reversed if . The special cases of inequality (1) are the subject of the following theorem, which is also due to Hardy et al. [1, Theorem 330].

Theorem A. *Let be nonnegative and Lebesgue integrable on or for every , according to or . Then
*

where The signs of the inequalities are reversed if .

As is well known, inequalities (2) play a very important role in both theory and applications. Ever since Hardy discovered inequalities (2), they have been studied by many authors, who either reproved them using various techniques or improved, generalized, and applied them in many different ways (see e.g. [2–22] and references therein). For further remarks concerning the improvements and properties of inequalities (2) and their generalizations, see for example, [23] or [24].

In the year 1977, Imoru [6] obtained the following integral inequalities which are related to Hardy's (see Theorem A).

Theorem B. *Let be continuous and nondecreasing on with , for and . Let be nonnegative and Lebesgue integrable with respect to on or on according to or , where , . Suppose
**
If , then
**
with both signs of inequalities reversed if .*

Later, in 1981, Chan in [2] derived several exponential generalizations of the Imoru's inequalities (5). In 1985, Imoru in [7] presented further extensions of (5). Moreover, in 1988, Yang et al. [22] gave some new generalizations of (5). Recently, Oguntuase and Imoru in [10] obtained other generalizations of the Yang et al.'s results.

The main purpose of this work is to give some improvements of inequalities (5) by using Hu Ke's inequality which is a sharp Hölder's inequality.

#### 2. A Set of Lemmas

In this section, we will prove lemmas, which play crucial roles in proving our main results.

Lemma 1 (see [25] Hu Ke's inequality). *Let , and be integrable functions defined on and for all , and let . Then
**
where .*

Lemma 2 (see [26]). *Let , and be integrable functions defined on and for all , let , and let . Then
**
where .*

Lemma 3. *Let be continuous and nondecreasing on . Let and be integrable functions and , for all , and let be nondecreasing. If , then
**
where . If , then
**
where .*

*Proof. *From Lemmas 1 and 2, the conclusion is easy to obtain.

Lemma 4. *Let be continuous and nondecreasing on with , for and . Let , , and , be nonnegative and Lebesgue integrable with respect to on or on according to or , where , , and let for all , . Suppose
**
If , then *

where is as in Theorem B, . If , then

where is as in Theorem B, .

*Proof. *We only prove inequality (12); the proofs of (13) and (14) are similar. Let , in inequality (8). Then, if , , we have

This proves inequality (12). Lemma 4 is proved.

Lemma 5. *With notation as in Lemma 4, one has the results as follows. If , then
**
If , then
*

*Proof. *We only prove inequality (16); the proofs of (17) and (18) are similar. If , , by using inequality (8), we have
where , . This proves inequality (16). Lemma 5 is proved.

Lemma 6 (see [23]). *If , , or , then
**
The inequality is reversed for . *

#### 3. Refinements of Hardy-Type Inequalities

Theorem 7. *Let be continuous and nondecreasing on with , for and . Let , and be nonnegative and Lebesgue integrable with respect to on or on according to or , where , , and let for all , . Suppose is as in Theorem B. If , then
**
where , ,
**
If , then
**
where , ,
*

*Proof. *We only prove the case ; the proof of case is similar.(i) When , by using the nondecreasing property of , we have
and hence
from which and from inequality (12) we have, on using integration by parts,

That is,

Combining inequalities (16), (20), and (29) yields inequality (21).(ii) When , by the same method as in case (i), we obtain and hence from which and from inequality (13) we have, on using integration by parts,

That is,

Combining inequalities (17), (20), and (33) yields inequality (22). The proof of Theorem 7 is complete.

#### Acknowledgments

The authors would like to express hearty thanks to the anonymous referees for their great efforts to improve this paper. This work was supported by the NNSF of China (Grant no. 61073121) and the Fundamental Research Funds for the Central Universities (Grant no. 13ZD19).

#### References

- G. H. Hardy, J. E. Littlewood, and G. Pólya,
*Inequalities*, Cambridge University Press, Cambridge, UK, 2nd edition, 1952. View at MathSciNet - L.-Y. Chan, “Some further extensions of Hardy's inequality,”
*Canadian Mathematical Bulletin*, vol. 24, no. 4, pp. 393–400, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Čižmešija, J. Pečarić, and L.-E. Persson, “On strengthened Hardy and Pólya-Knopp's inequalities,”
*Journal of Approximation Theory*, vol. 125, no. 1, pp. 74–84, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Deng, S. Wu, and D. He, “A sharpened version of Hardy's inequality for parameter $p=5/4$,”
*Journal of Inequalities and Applications*, vol. 2013, p. 63, 2013. View at Publisher · View at Google Scholar - K. Hedayatian and L. Karimi, “On convexity of composition and multiplication operators on weighted Hardy spaces,”
*Abstract and Applied Analysis*, vol. 2009, Article ID 931020, 9 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. O. Imoru, “On some integral inequalites related to Hardy's,”
*Canadian Mathematical Bulletin*, vol. 20, no. 3, pp. 307–312, 1977. View at Google Scholar · View at MathSciNet - C. O. Imoru, “On some extensions of Hardy's inequality,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 8, no. 1, pp. 165–171, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Krnić, J. Pečarić, and P. Vuković, “On some higher-dimensional Hilbert's and Hardy-Hilbert's integral inequalities with parameters,”
*Mathematical Inequalities & Applications*, vol. 11, no. 4, pp. 701–716, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Krulić, J. Pečarić, and L.-E. Persson, “Some new Hardy type inequalities with general kernels,”
*Mathematical Inequalities & Applications*, vol. 12, no. 3, pp. 473–485, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. A. Oguntuase and C. O. Imoru, “New generalizations of Hardy's integral inequality,”
*Journal of Mathematical Analysis and Applications*, vol. 241, no. 1, pp. 73–82, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. T. Shum, “On integral inequalities related to Hardy's,”
*Canadian Mathematical Bulletin. Bulletin Canadien de Mathématiques*, vol. 14, pp. 225–230, 1971. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Tian, “Extension of Hu Ke's inequality and its applications,”
*Journal of Inequalities and Applications*, vol. 2011, p. 77, 2011. View at Google Scholar - J. Tian, “Inequalities and mathematical properties of uncertain variables,”
*Fuzzy Optimization and Decision Making*, vol. 10, no. 4, pp. 357–368, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Tian, “Property of a Hölder-type inequality and its application,”
*Mathematical Inequalities and Applications*, vol. 16, no. 3, pp. 831–841, 2013. View at Google Scholar - J. Tian and X. M. Hu, “A new reversed version of a generalized sharp Hölder's inequality and its applications,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 901824, 9 pages, 2013. View at Publisher · View at Google Scholar - J. Tian and X. M. Hu, “Refinements of generalized Hölder's inequality,”
*Journal of Mathematical Inequalities*. In press. - J. Tian and S. Wang, “Refinements of generalized Aczel's inequality and Bellman's inequality and their applications,”
*Journal of Applied Mathematics*, vol. 2013, Article ID 645263, 6 pages, 2013. View at Publisher · View at Google Scholar - S. H. Wu, “Generalization of a sharp Hölder's inequality and its application,”
*Journal of Mathematical Analysis and Applications*, vol. 332, no. 1, pp. 741–750, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Wu, “A new sharpened and generalized version of Hölder's inequality and its applications,”
*Applied Mathematics and Computation*, vol. 197, no. 2, pp. 708–714, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Wu, “On the weighted generalization of the Hermite-Hadamard inequality and its applications,”
*The Rocky Mountain Journal of Mathematics*, vol. 39, no. 5, pp. 1741–1749, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Wu and L. Debnath, “Generalizations of Aczél's inequality and Popoviciu's inequality,”
*Indian Journal of Pure and Applied Mathematics*, vol. 36, no. 2, pp. 49–62, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Yang, Z. Zeng, and L. Debnath, “On new generalizations of Hardy's integral inequality,”
*Journal of Mathematical Analysis and Applications*, vol. 217, no. 1, pp. 321–327, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. F. Beckenbach and R. Bellman,
*Inequalities*, Springer, New York, NY, USA, 1983. View at Zentralblatt MATH · View at MathSciNet - D. S. Mitrinović, J. E. Pečarić, and A. M. Fink,
*Classical and New Inequalities in Analysis*, Kluwer Academic, Dordrecht, The Netherlands, 1993. View at MathSciNet - K. Hu, “On an inequality and its applications,”
*Scientia Sinica. Zhongguo Kexue*, vol. 24, no. 8, pp. 1047–1055, 1981. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Tian, “Reversed version of a generalized sharp Hölder's inequality and its applications,”
*Information Sciences*, vol. 201, pp. 61–69, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet