Journal of Applied Mathematics
Volume 2014 (2014), Article ID 274310, 9 pages
http://dx.doi.org/10.1155/2014/274310

Research Article

Refinements of Generalized Hölder’s Inequalities

Jingfeng Tian,¹ Wen-Li Wang,² and Donal O’Regan3,4

¹College of Science and Technology, North China Electric Power University, Baoding 071051, China
²Department of Information Engineering, China University of Geosciences, Baoding 071000, China
³Department of Mathematics, National University of Ireland, Galway, Ireland
⁴NAAM-Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 4 June 2014; Accepted 13 November 2014; Published 27 November 2014

Academic Editor: Engang Tian

Copyright © 2014 Jingfeng Tian et al. 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.

We present some new versions of generalized Hölder’s inequalities. The results are used to improve Minkowski’s inequality and a Beckenbach-type inequality.

1. Introduction

If ,  ,  ,  , then

The sign of the inequality is reversed if , (for , we assume that ). Inequality (1) and its reversed version are called Hölder’s inequality.

In 1979, Vasić and Pečarić [1] presented the following result.

Theorem A. Let ,  .

(a) If and if , then

(b)If , then

(c)If ,  , and if , then

Inequalities (2), (3), and (4) are called generalized Hölder’s inequalities. It is well known that Hölder’s inequality and generalized Hölder’s inequalities are important in mathematical analysis and in the field of applied mathematics. For example, Agahi et al. [2] presented generalizations of the Hölder and the Minkowski inequality for pseudointegrals and Liu [3] established a Hölder type inequality. For a discussion on inequalities we refer the reader to [1, 4–9] and the references therein. Although generalized Hölder’s inequalities play an important and basic role in many branches of mathematics, some problems can not be precisely estimated by generalized Hölder’s inequalities. For example, if we set , , , , , , , , ,  , then from generalized Hölder’s inequality (2) we obtain . It is of interest to develop a refinement of Hölder’s inequality.

In this paper we present new refinements of inequalities (2), (3), and (4) in Section 2. In Section 3, we use our results to improve the Minkowski inequality and a Beckenbach-type inequality.

2. Refinements of Generalized Hölder’s Inequalities

We begin with a known result.

Lemma 1 (see [10]). If , , or , then

The inequality is reversed for .

Lemma 2. Let and .

(a)If and if , then

(b)If , then

(c)If , , and if , then

Proof. (a) Note first that and ().
Case ( I). Let be even.
Note that   +    +  , and, using inequality (2), we have

so (6) holds when is even.
Case (II). Let be odd.
Note that   +    +  , and, using inequality (2), we have

so (6) holds for is odd.
(b) Using similar reasoning as in Case (a) and using inequality (3), we obtain inequality (7).
(c) The proof of inequality (8) is similar to the reasoning used to prove inequality (6) so we omit it.
The proof of Lemma 2 is complete.

Next, we present new refinements of inequalities (2), (3), and (4).

Theorem 3. Let ,  , and let be any given natural number .

(a)If and if , then

(b)If , , and if , then

(c)If , then

Proof. (a) Consider the substitution

It is easy to see that, for any given natural number , the following inequalities hold:

Consequently, by using substitution (14) and inequality (6), we have

and thus we have

that is,

We have the desired inequality (11). The proof of inequalities (12) and (13) is similar to the reasoning used to prove inequality (11) so we omit the proof.

From Theorem 3, we obtain the following new refinements of generalized Hölder inequalities (2), (3), and (4).

Theorem 4. Let ,  , and let be any given natural number .

(a)If and if , then

(b)If , , and if , then

(c)If , then

From Lemma 1 and Theorem 4, we obtain the following refinements of Hölder’s inequality.

Theorem 5. Let ,  , and let be any given natural number .

(a)If and if , then

(b)If , , and if , then

(c)If , then

In particular, putting ,  ,  , ,   in inequality (19) and putting ,  ,  , ,   in inequalities (20) and (21), respectively, we obtain the following corollary.

Corollary 6. Let , and let be any given natural number .

(a)If ,  , then

(b)If ,  ,  , then

(c)If , then

Remark 7. Let ,  ,  ,  ,  ,  , and  , and let ,  . Then from inequality (25) we obtain .

Similarly, putting ,  ,  ,  ,   in inequality (22) and putting ,  ,  ,  ,   in inequalities (23) and (24), respectively, we obtain the following corollary.

Corollary 8. Let , and let be any given natural number .

(a)If , , then

(b)If ,  ,  , then

(c)If , then

3. Applications

In this section, we give two applications of our new inequalities. Firstly, we present a refinement of Minkowski’s inequality.

Theorem 9. Let ,  , and let be any given natural number . If , then

If , then

Proof. Consider the following.
Case (i). Let .
Now

and apply Corollary 6 with indices and to each sum on the right so

From Lemma 1 we have

Dividing both sides by , we obtain the desired inequality.
Case (ii). Let .
Similar reasoning as in Case (i) yields inequality (32).

Now, we give a sharpened version of a Beckenbach-type inequality. The Beckenbach inequality [11] was generalized and extended in several directions (see, e.g., [12–15]). In 1983, Wang [16] presented the following Beckenbach-type inequality.

Theorem B. Let ,  , let be positive numbers, and let , be positive integrable functions defined on . Then

where . The sign of the inequality in (36) is reversed if .

From Corollary 8, we obtain a new refinement of Beckenbach-type inequality (36).

Theorem 10. Let be positive numbers, and let , be positive integrable functions defined on . If ,  , then

where .
If ,  ,  , then

where .

Proof. We first consider the case ,  . Using simple computations, we have

Moreover, from Corollary 8 we obtain

that is,

Then combining inequalities (39) and (41) yields inequality (37).
Using the above reasoning and applying inequality (26), it is easy to obtain inequality (38).

4. Conclusions

In this paper we presented some new refinements of Hölder’s inequality and we obtained a refinement of Cauchy’s inequality. We improved Minkowski inequality and a Beckenbach-type inequality. In future research we hope to obtain new results using inequalities (11), (12), (13), and (19)–(30).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (no. 13ZD19) and the Higher School Science Research of Hebei Province of China (no. Z2013038).

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