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