/ / Article

Research Article | Open Access

Volume 2013 |Article ID 645263 | 6 pages | https://doi.org/10.1155/2013/645263

# Refinements of Generalized Aczél's Inequality and Bellman's Inequality and Their Applications

Accepted30 Dec 2012
Published04 Feb 2013

#### 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  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  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 . The case was given in  by Vasić and Pečarić.

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

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

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

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

In 2012, Tian  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 , 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  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 . The case was proved in  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).

1. J. Aczél, “Some general methods in the theory of functional equations in one variable. New applications of functional equations,” Uspekhi Matematicheskikh Nauk, vol. 11, no. 3, pp. 3–68, 1956 (Russian). View at: Google Scholar | MathSciNet
2. E. F. Beckenbach and R. Bellman, Inequalities, Springer, Berlin, Germany, 1983. View at: MathSciNet
3. J. L. Díaz-Barrero, M. Grau-Sánchez, and P. G. Popescu, “Refinements of Aczél, Popoviciu and Bellman's inequalities,” Computers & Mathematics with Applications, vol. 56, no. 9, pp. 2356–2359, 2008.
4. G. Farid, J. Pečarić, and A. Ur Rehman, “On refinements of Aczél, Popoviciu, Bellman's inequalities and related results,” Journal of Inequalities and Applications, vol. 2010, Article ID 579567, 17 pages, 2010.
5. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 1952. View at: MathSciNet
6. D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, vol. 61, Kluwer Academic, Dordrecht, The Netherlands, 1993. View at: MathSciNet
7. Y. Ouyang and R. Mesiar, “On the Chebyshev type inequality for seminormed fuzzy integral,” Applied Mathematics Letters, vol. 22, no. 12, pp. 1810–1815, 2009.
8. T. Popoviciu, “On an inequality,” Gazeta Matematica şi Fizica A, vol. 11, no. 64, pp. 451–461, 1959 (Romanian).
9. J. Tian, “Inequalities and mathematical properties of uncertain variables,” Fuzzy Optimization and Decision Making, vol. 10, no. 4, pp. 357–368, 2011. View at: Publisher Site | Google Scholar | MathSciNet
10. J. Tian, “Reversed version of a generalized Aczél's inequality and its application,” Journal of Inequalities and Applications, vol. 2012, article 202, 2012. View at: Google Scholar
11. J. Tian, “Reversed version of a generalized sharp Hölder's inequality and its applications,” Information Sciences, vol. 201, pp. 61–69, 2012. View at: Publisher Site | Google Scholar | MathSciNet
12. J. Tian, “Extension of Hu Ke's inequality and its applications,” Journal of Inequalities and Applications, vol. 2011, article 77, 2011. View at: Google Scholar
13. J. Tian, “Property of a Hölder-type inequality and its application,” Mathematical Inequalities & Applications. In press. View at: Google Scholar
14. J. Tian and X. M. Hu, “A new reversed version of a generalized sharp Hölder's inequality and its applications,” Abstract and Applied Analysis. In press. View at: Google Scholar
15. P. M. Vasić and J. E. Pečarić, “On the Hölder and some related inequalities,” Mathematica, vol. 25, no. 1, pp. 95–103, 1982.
16. P. M. Vasić and J. E. Pečarić, “On the Jensen inequality for monotone functions,” Analele Universitatii din Timişoara, vol. 17, no. 1, pp. 95–104, 1979.
17. S. Vong, “On a generalization of Aczél's inequality,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1301–1307, 2011.
18. S. Wu and L. Debnath, “Generalizations of Aczél's inequality and Popoviciu's inequality,” Indian Journal of Pure and Applied Mathematics, vol. 36, no. 2, pp. 49–62, 2005.
19. W. Yang, “Refinements of generalized Aczél-Popoviciu's inequality and Bellman's inequality,” Computers & Mathematics with Applications, vol. 59, no. 11, pp. 3570–3577, 2010.
20. R. Bellman, “On an inequality concerning an indefinite form,” The American Mathematical Monthly, vol. 63, pp. 108–109, 1956.