#### 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).