#### Abstract

We give some refinements of generalized Aczél's inequality and Bellman's inequality proposed by Tian. As applications, some refinements of integral type of generalized Aczél's inequality and Bellman's inequality are given.

#### 1. Introduction

The famous Aczél’s inequality [1] states that if , are real numbers such that and , then

It is well known that Aczél’s inequality plays an important role in the theory of functional equations in non-Euclidean geometry. In recent years, various attempts have been made by many authors to improve and generalize the Aczél’s inequality (see [2–19] and references therein). We state here some improvements of Aczél’s inequality.

One of the most important results in the references mentioned above is an exponential extension of (1), which is stated in the following theorem.

Theorem A. *Let and be real numbers such that , , and and let , be positive numbers such that and . If , then
**
If , then the reverse inequality in (2) holds.*

*Remark 1. *The case of Theorem A was proved by Popoviciu [8]. The case was given in [15] by Vasić and Pečarić.

Vasić and Pečarić [16] presented a further extension of inequality (1).

Theorem B. *Let , , , , , and let . Then
*

In a recent paper [18], Wu and Debnath established an interesting generalization of Aczél’s inequality [1] as follows.

Theorem C. *Let , , , , , and let . Then
*

In 2012, Tian [10] presented the following reversed version of inequality (4).

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

Therefore, applying the above inequality, Tian gave the reversed version of inequality (3) as follows.

Theorem E. *Let , , , , , and , . Then
*

Moreover, in [10], Tian obtained the following integral form of inequality (5).

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

Bellman inequality [20] related with Aczél’s inequality is stated as follows.

Theorem G. *Let , be positive numbers such that and . If , then
**
If , then the reverse inequality in (8) holds.*

*Remark 2. *The case of Theorem G was proposed by Bellman [20]. The case was proved in [15] by Vasić and Pečarić.

The main purpose of this work is to give refinements of inequalities (5) and (8). As applications, some refinements of integral type of inequality (5) and (8) are given.

#### 2. Refinements of Generalized Aczl’s Inequality and Bellman’s Inequality

Theorem 3. *Let , , , , , , and let . Then
**
where
*

*Proof**Case* (I). When , then . On the one hand, we split the left-hand side of inequality (9) as follows:
where
From this hypothesis, it is immediate to obtain the inequality
Thus, by using inequality (6), we have

On the other hand, by using inequality (6) again, we obtain the inequality
Combining inequalities (11), (14), and (15) we can get inequality (9).*Case* (II). When , using Case (I) with , we have
where , , , .

Hence, taking , into (16), we obtain
that is,
Therefore, repeating the foregoing arguments, we get
Combining inequalities (18) and (19) leads to inequality (9) immediately. The proof of Theorem 3 is completed.

If we set , then from Theorem 3, we obtain the following refinement of inequality (6).

Corollary 4. *Let , , , , , , and . Then
*

Putting , , , , and in Theorem 3, we obtain the refinement and generalization of Theorem A for .

Corollary 5. *Let , , , , , , and . Then, the following inequality holds:
*

Based on the mathematical induction, it is easy to see that the following generalized Bellman’s inequality is true.

Theorem 6. *Let , , , , and let . Then
*

Next, we give a refinement of generalized Bellman’s inequality (22) as follows.

Theorem 7. *Let , , , , and let . Then
*

*Proof. *The proof of Theorem 7 is similar to the one of Theorem 3. Applying generalized Bellman’s inequality (22) twice, we can deduce the inequality (23).

#### 3. Application

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

Firstly, we give an improvement of inequality (7) by using Theorem 3.

Theorem 8. *Let , let , , , and let be positive integrable functions defined on with . Then, for any , one has
*

*Proof. *We need to prove only the left side of inequality (24). The proof of the right side of inequality (24) is similar. For any positive integers and , we choose an equidistant partition of and , respectively, as
Noting that , we have
Consequently, there exists a positive integer , such that
for all and .

By using Theorem 3, for any , the following inequality holds:

Since

we have

Noting that are positive Riemann integrable functions on , we know that and are also integrable on . Letting on both sides of inequality (30), we get the left side of inequality (24). The proof of Theorem 8 is completed.

We give here a direct consequence from Theorem 8. Putting , , , , , , and in (24), we obtain a special important case as follows.

Corollary 9. *Let and be real numbers such that , , and , let , and let be positive integrable functions defined on with and . Then, for any , one has
*

Finally, we present a refinement of integral type of generalized Bellman’s inequality.

Theorem 10. *Let , let , and let be positive integrable functions defined on with . Then, for any , one has
*

*Proof. *The proof of Theorem 10 is similar to the proof of Theorem 8.

A special case to the last theorem is as follows.

Corollary 11. *Let , let , and let be positive integrable functions defined on with and . Then, for any , one has
*

#### Acknowledgments

The authors would like to express hearty thanks to the anonymous referees for their great efforts to improve this paper. This work was supported by the NNSF of China (no. 61073121), the Natural Science Foundation of Hebei Province of China (no. F2012402037), the Natural Science Foundation of Hebei Education Department (no. Q2012046), and the Fundamental Research Funds for the Central Universities (no. 11ML65).