Abstract

We present several new generalized versions of refined Hölder’s inequalities proposed by Tian and Hu. And then we obtained some new generalized and sharp versions of Hölder’s inequalities. As the applications, the obtained results are used to improve Aczél-Popoviciu type inequality and Aczél-Vasić-Pečarić inequality.

1. Introduction

The famous Hölder’s inequality asserts that where , and . The sign of inequality is reversed for . For , we assume that . The above inequality plays an important role in many areas of pure and applied mathematics. In recent years, considerable attention has been given to this inequality involving its generalizations, refinements, variations, and applications (see [1–5] and references therein). Among various generalizations of (1), Vasić and Pečarić in [6] presented the following important theorems.

Theorem A. Let . If are positive numbers such that , then

Theorem B. Let .(a)If , and if , then (b)If ,

In 2013, Tian and Hu [7] established some interesting refinements of inequalities (2), (3), and (4) as follows.

Theorem C. Let , let , , and let Then

Theorem D. Let .(a)If , and if , then (b)If , and if , then (c)If , then where .

The main goal of this paper is to give generalizations of inequality (5), (6), (7), and (8), and then some new generalized and sharp versions of Hölder’s inequalities are given. Moreover, the obtained results will be applied to improve Aczél-Popoviciu type inequality and Aczél-Vasić-Pečarić inequality.

2. Generalizations of Refined Hölder’s Inequalities

In order to prove our main results, we need the following lemmas.

Lemma 1 (see [8]). If or , then The sign of inequality is reversed for .

Lemma 2 (see [9]). If , then The sign of inequality is reversed for or .

Lemma 3 (see [10]). Let . Then

Next we provide the generalization of inequality (5).

Theorem 4. Let , let , and let , . Then

Proof. After some simple calculations, we obtain Case (I). When , then . Obviously, inequality (12) is equivalent to inequality (5).
Case (II). Let be even and let . Write , which implies . By inequality (2), we have Consequently, according to      , by using inequality (2) on the right side of (14), we observe that Additionally, by using Lemma 2 on the right side of (15), we have Combining inequalities (13), (15), and (16) we can get
Furthermore, noting that we have Consequently, from (17) and (19), we obtain the desired inequality (12) when is even.
Case (III). Let be odd and let . By the same method as in the above Case , we have

If we set , then from Theorem 4 we obtain a new generalized and sharp version of Hölder’s inequality (1) as follows.

Corollary 5. Let , and let . Then

Moreover, by using Theorem 4 and Lemma 3 we obtain the following refinement and generalization of the Hölder’s inequality (2).

Corollary 6. Let , let , and let . Then

Similarly, if we set , , , , , then from Corollary 6 we obtain the following refinement and generalization of the Hölder’s inequality (1).

Corollary 7. Let , , and let . Then

Next, we will give the generalizations of inequalities (6), (7), and (8).

Theorem 8. Let , let , , and let . Then where .

Proof. Case (I). When , with . Obviously, , which implies . From inequality (3), we have
Consequently, according to     , by using the inequality (3) on the right side of (25), we observe that Moreover, using Lemma 2 together with , we find
Combining inequalities (13), (26), and (27) we can get
Furthermore, noting that we have Consequently, from Lemma 1 and the inequalities (28) and (30), we have the desired inequality (24).
Case (II). When with , then . The inequality (24) is equivalent to inequalities (6) and (7).
Case (III). When , then . Inequality (24) is equivalent to inequality (8).

If we set , then from Theorem 8 we obtain the following refinement and generalization of the Hölder’s inequality (1).

Corollary 9. Let , and let . Then where .

3. Applications

In this section, we show two applications of our new inequalities. Firstly, we provide an application of the obtained results to improve the Aczél-Popoviciu type inequality, which is due to Wu and Debnath.

The classical Aczél [11] inequality states that if , , , , then

As is well-known, Aczél’s inequality plays an important role in the theory of functional equations in non-Euclidean geometry, and many researchers (see [12–17] and references therein) have given considerable attention to this inequality and its generalizations and refinements.

In 1959, Popoviciu [18] presented the following extension of inequality (32).

Theorem E. Let , and let ,  ,  ,  . Then

The inequality (33) is called Aczél-Popoviciu inequality.

Later, in 1982, Vasić and Pečarić [5] obtained the reversed version of inequality (33). The inequality is called Aczél-Vasić-Pečarić inequality.

Theorem F. Let , and let , , , . Then

In a recent paper [19], Wu and Debnath established an interesting generalization of Aczél-Popoviciu inequality (33) as follows.

Theorem G. Let , , , , , , and let . Then

Next, by using Corollary 6 we present the following refinement of inequality (35).

Theorem 10. Let , let , ,   , , , and let . Then where .

Proof. Denote Since , we obtain . From Corollary 6, we have therefore, Substituting into the inequality (39), we have the desired inequality (36).

Finally, we give the following Aczél-Vasić-Pečarić type inequality, which is the refinement of Aczél-Vasić-Pečarić inequality (34).

Theorem 11. Let , let , , ,  , and let . Then where .

Proof. By the same method as in Theorem 10, from Theorem 8, we get the desired inequality (41).

Conflict of Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors would like to express their gratitude to the referee for his/her very valuable comments and suggestions. This work was supported by the NNSF of China (no. 61073121) and the Fundamental Research Funds for the Central Universities (no. 13ZD19).