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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 727923, 8 pages
http://dx.doi.org/10.1155/2013/727923
Research Article

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. [222] 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).

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