Journal of Applied Mathematics

Volume 2014 (2014), Article ID 581354, 7 pages

http://dx.doi.org/10.1155/2014/581354

Research Article

Refinements of Aczél-Type Inequality and Their Applications

Jingfeng Tian¹ and Wen-Li Wang²

¹College of Science and Technology, North China Electric Power University, Baoding, Hebei 071051, China

²Department of Information Engineering, China University of Geosciences Great Wall College, Baoding 071000, China

Received 22 April 2014; Revised 8 June 2014; Accepted 9 June 2014; Published 26 June 2014

Academic Editor: Shanhe Wu

Copyright © 2014 Jingfeng Tian and Wen-Li Wang. 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 sharpened versions of Aczél-type inequality. Moreover, as applications, some refinements of integral type of Aczél-type inequality are given.

1. Introduction

Let be a positive integer, and let , be real numbers such that or . Then, the famous Aczél inequality [1] can be stated as follows:

Aczél’s inequality plays a very important role in the theory of functional equations in non-Euclidean geometry. Due to the importance of Aczél’s inequality (1), it has received considerable attention by many authors and has motivated a large number of research papers giving it various generalizations, improvements, and applications (see [2–21] and the references therein).

In 1959, Popoviciu [10] first obtained an exponential extension of the Aczél inequality as follows.

Theorem B. Let , , and let , be positive numbers such that and . Then

Later, in 1982, Vasić and Pečarić [16] established the following reversed version of inequality (2).

Theorem C. Let , , , and let , be positive numbers such that and . Then

In another paper, Vasić and Pečarić [15] generalized inequality (2) in the following form.

Theorem D. Let , , , , , and let . Then

In 2012, Tian [13] presented the reversed version of inequality (4) as follows.

Theorem E. Let , , , , , , . Then

Moreover, in [13] Tian established an integral type of inequality (5).

Theorem F. Let , , , let , and let be positive Riemann integrable functions on such that . Then

Remark 1. In fact, the integral form of inequality (4) is also valid; that is, one has the following.

Theorem G. Let , , let , and let be positive Riemann integrable functions on such that . Then

The main purpose of this work is to give new refinements of inequalities (4) and (5). As applications, new refinements of inequalities (6) and (7) are also given.

2. Refinements of Aczél-Type Inequality

In order to present our main results, we need some lemmas as follows.

Lemma 2 (see [6]). Let be real numbers such that and . If , then

If either or and if all are positive or negative with , then the reverse inequality of (8) holds.

Lemma 3 (see [15]). Let .

(a)If and if , then

(b)If , then

(c)If , , and , then

Lemma 4 (see [18]). Let , . Then

Lemma 5. Let , , , let , and let .

Then

Proof. From the assumptions we have that

Case (I) (let be even). In view of by using inequality (9),

we get

On the other hand, applying Lemma 4 and the arithmetic-geometric means inequality we obtain

Applying Lemma 4 again, we get

Combining (15), (16), and (17) yields immediately inequality (13).

Case (II) (let be odd). In view of ++++++=++, by using inequality (9), we have

On the other hand, applying Lemma 4 and the arithmetic-geometric means inequality we obtain

Applying Lemma 4 again, we have

Hence, combining (18), (19), and (20) yields immediately inequality (13).

Similar to the proof of Lemma 5 but using Lemma 2 in place of Lemma 4, we immediately obtain the following result.

Lemma 6. Let , , , , let , , and let .

Then

Using the same methods as in Lemma 6, we get the following Lemma.

Lemma 7. Let , , let , and let .

Then

Now, we present some new refinements of inequalities (4) and (5).

Theorem 8. Let , , , , let , , , and let .

Then

Proof. From the assumptions we find that

Thus, by using Lemma 5 with a substitution in (13), we obtain

which implies

On the other hand, we get from Lemma 3 that

Combining (26) and (27) yields immediately the desired inequality (23).

Theorem 9. Let , , , , , let , and let .

Then

Inequality (28) is also valid for , , .

Proof. The proof of Theorem 9 is similar to the one of Theorem 8, and we omit it.

3. Applications

In this section, we show two applications of the inequalities newly obtained in Section 2.

Firstly, we present a new refinement of inequality (6) by using Theorem 9.

Theorem 10. Let , , , , let be positive integrable functions defined on with , and let .

Then

Proof. For any positive integer , we choose an equidistant partition of as

Since , it follows that

Therefore, there exists a positive integer such that

for all and .

Moreover, for any , it follows from Theorem 9 that

Noting that

we get

In view of the assumption that are positive Riemann integrable functions on , we find that and are also integrable on . Letting on both sides of inequality (36), we get the desired inequality (29).

Next, we give a new refinement of inequality (7) by using Theorem 8.

Theorem 11. Let , , , , and let be positive integrable functions defined on with , and let .

Then

Proof. The proof of Theorem 11 is similar to the one of Theorem 10, and we omit it.

Conflict of Interests

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

Acknowledgments

The authors would like to thank the reviewers and the editors for their valuable suggestions and comments. This work was supported by the Fundamental Research Funds for the Central Universities (Grant no. 13ZD19).

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