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New refinements of generalized Aczél inequality
Journal of Inequalities and Applications volume 2014, Article number: 239 (2014)
Abstract
In this article, we present several new refinements of the generalized Aczél inequality. As an application, an integral type of the generalized Aczél-Vasić-Pečarić inequality is refined.
MSC:26D15, 26D10.
1 Introduction
In 1956, Aczél [1] established the following inequality, which is called the Aczél inequality.
Theorem A Let , (), , . Then
As is well known, the Aczél inequality plays an important role in the theory of functional equations in non-Euclidean geometry, and many authors (see [2–6] and references therein) have given considerable attention to this inequality and its refinements.
In 1959, Popoviciu [3] generalized the Aczél inequality (1) in the form asserted by Theorem B below.
Theorem B Let , , , let , (), , . Then
Later, in 1982, Vasić and Pečarić [7] presented the reversed version of inequality (2), which is stated in the following theorem. The inequality is called the Aczél-Vasić-Pečarić inequality.
Theorem C Let (), , and let , (), , . Then
In another paper, Vasić and Pečarić [8] presented an interesting generalization of inequality (2). The inequality is called the generalized Aczél-Vasić-Pečarić inequality.
Theorem D Let , , , , , and let . Then
In 2012, Tian [5] gave the reversed version of inequality (4) in the following form.
Theorem E Let , (), , and let , , , . Then
Moreover, in [5] Tian established an integral type of generalized Aczél-Vasić-Pečarić inequality.
Theorem F Let , (), , let (), and let () be positive Riemann integrable functions on such that . Then
The main object of this paper is to give several new refinements of inequality (4) and (5). As an application, a new refinement of inequality (6) is given.
2 New refinements of generalized Aczél inequality
In order to prove the main results in this section, we need the following lemmas.
Lemma 2.1 [5]
Let (, ), let be a real number, (), and let . Then
Lemma 2.2 [9]
Let (, ), let (), and let . Then
Lemma 2.3 [10]
If , or , then
The inequality is reversed for .
Lemma 2.4 [10]
Let be real numbers, let m be a natural number, and let . Then
Lemma 2.5 Let , let (), and let . Then
Proof From the assumptions in Lemma 2.5, we find
and
Thus, by using inequality (7) we have
Noting the fact that there are product terms in the expression , and using the arithmetic-geometric mean’s inequality, we obtain
Therefore, we have
On the other hand, from Lemma 2.4 we have
Consequently, from (13), (15), and (16), we obtain the desired inequality (11). □
Lemma 2.6 Let , , let , (), and let . If , then
If , then
Proof Case I. When . Let us consider the following product:
From the hypotheses of Lemma 2.6, it is easy to see that
and
Then, applying inequality (7), we have
There are product terms in the expression , and then we derive from the arithmetic-geometric mean’s inequality that
Therefore, we have
On the other hand, from Lemma 2.4 we find
Combining inequalities (21), (23), and (24) yields the desired inequality (17).
Case II. When . By the same method as in Lemma 2.5, it is easy to obtain the desired inequality (18). So we omit the proof. The proof of Lemma 2.6 is completed. □
Lemma 2.7 Let , let (), and let , . Then
Proof By the same method as in Lemma 2.5, applying Lemma 2.2, it is easy to obtain the desired inequality (25). So we omit the proof. □
Lemma 2.8 Let , let (), and let . Then
Proof After simply rearranging, we write by the component of in increasing order, where is a permutation of .
Then from Lemma 2.5 and Lemma 2.4 we get
The proof of Lemma 2.8 is completed. □
By the same method as in Lemma 2.8, we obtain the following two lemmas.
Lemma 2.9 Let , , let , (), and let . If , then
If , then
Lemma 2.10 Let , let (), and let , . Then
Now, we give the refinement and generalization of inequality (5).
Theorem 2.11 Let , , , , , and let . Then
Proof From the assumptions in Theorem 2.11, it is easy to verify that
It thus follows from Lemma 2.8 with the substitution in (26) that
which implies
On the other hand, it follows from Lemma 2.1 that
Combining inequalities (34) and (35) yields inequality (31).
The proof of Theorem 2.11 is completed. □
Theorem 2.12 Let , (), let , , , , and let . If , then
If , then
Proof From the hypotheses of Theorem 2.12, we find that
and
Consequently, by the same method as in Theorem 2.11, and using Lemma 2.9 with a substitution () in (28) and (29), respectively, we obtain the desired inequalities (36) and (37). □
By the same method as in Theorem 2.11, and using Lemma 2.10, we obtain the following sharpened and generalized version of inequality (4).
Theorem 2.13 Let , , , , , let , and let . Then
Therefore, from Lemma 2.3 and Theorem 2.13 we get a new refinement and generalization of inequality (4).
Corollary 2.14 Let , , , , , let , and let . If , then
If , then
Remark 2.15 If we set in Corollary 2.14, then inequalities (39) and (40) reduce to Wu’s inequality ([[11], Theorem 1]).
In particular, putting , , , , () in Theorem 2.13, we obtain a new refinement and generalization of inequality (2).
Corollary 2.16 Let , (), let , , and let , . Then
Similarly, putting , , , , () in Theorem 2.12 and Theorem 2.11, respectively, we obtain a new refinement and generalization of inequality (3).
Corollary 2.17 Let , (), let , , , and let , . Then
From Lemma 2.3 and Theorem 2.11 we obtain the following refinement of inequality (5).
Corollary 2.18 Let , , , , , and let . Then
Similarly, from Lemma 2.3 and Theorem 2.12 we obtain the following refinement and generalization of inequality (5).
Corollary 2.19 Let , (), let , , , , and let , . Then
If we set , then from Corollary 2.18 and Corollary 2.19 we obtain the following refinement of inequality (5).
Corollary 2.20 Let , (), , let , , , , and let . Then
3 Application
In this section, we show an application of the inequality newly obtained in Section 2.
Theorem 3.1 Let (), let , (), , , and let () be positive integrable functions defined on with . Then
Proof For any positive integers n, we choose an equidistant partition of as
Noting that (), we have
Consequently, there exists a positive integer N, such that
for all and .
By using Theorem 2.12, 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 (48), we get the desired inequality (46). The proof of Theorem 3.1 is completed. □
Remark 3.2 Obviously, inequality (46) is sharper than inequality (6).
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Acknowledgements
The authors would like to express their gratitude to the referee for his/her very valuable comments and suggestions. This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. 13ZD19).
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Tian, J., Sun, Y. New refinements of generalized Aczél inequality. J Inequal Appl 2014, 239 (2014). https://doi.org/10.1186/1029-242X-2014-239
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DOI: https://doi.org/10.1186/1029-242X-2014-239