Refinements of Generalized Hölder’s Inequalities

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).