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Reversed version of a generalized Aczél’s inequality and its application
Journal of Inequalities and Applications volume 2012, Article number: 202 (2012)
In this paper, we give a reversed version of a generalized Aczél’s inequality which is due to Wu and Debnath. As an application, an integral type of the reversed version of the Aczél-Vasić-Pečarić inequality is obtained.
In 1956, Aczél  established the following inequality which is of wide application.
Theorem A If , () are positive numbers such that or , then
It is well known that Aczél’s inequality (1) plays an important role in the theory of functional equations in non-Euclidean geometry. Various refinements, generalizations and applications of inequality (1) have appeared in literature (see, e.g., [2–12],  and the references therein).
One of the most important results in the works mentioned above is the exponential generalization of (1) asserted by Theorem B.
Theorem B Let p and q be real numbers such that and , and let , () be positive numbers such that and . Then, for , we have
If (), we have the reverse inequality.
In another paper , Vasić and Pečarić presented the following extension of inequality (1).
Theorem C Let , , , , , and let . Then
Recently, it comes to our attention that an interesting generalization of Aczél’s inequality, which was established by Wu and Debnath in , is as follows.
Theorem D Let , , , , , and let . Then
and equality holds if and only if , for , or
The purpose of this work is to give a reversed version of inequality (4). As application, an integral type of the reversed version of the Aczél-Vasić-Pečarić inequality is obtained.
2 Reversed version of a generalized Aczél’s inequality
We need the following lemmas in our deduction.
Lemma 2.1 
If , , , , then
The inequality is reversed for or . In each case, the sign of the equality holds if and only if for all .
Lemma 2.2  (Generalized Hölder’s inequality)
Let (, ). If , (), , then
The sign of the equality holds if and only if the m sets are proportional.
Lemma 2.3 Let (, ), let , (), and let . Then
The sign of the equality holds if and only if the m sets are proportional for , or , for .
Proof Case (I). When , then . Obviously, inequality (7) is equivalent to inequality (6).
Case (II). When with . Write (), which implies . By inequality (6), we have
Consequently, according to , by using inequality (6) on the right side of (8), we observe that
Additionally, using Lemma 2.1 together with , we find
Combining inequalities (9) and (10) leads to inequality (7) immediately.
Case (III). When with . Obviously, inequality (7) is equivalent to inequality (6).
The condition of the equality for inequality can easily be obtained by Lemma 2.1 and Lemma 2.2. This completes the proof of Lemma 2.3. □
Remark 2.4 It is clear that the generalized Hölder inequality (6) is a simple consequence of Lemma 2.3 presented in this article.
Theorem 2.5 Let , , (), , , , and let . Then
and the equality holds if and only if , for , or
By using inequality (7), we have
Therefore, from (12), (13) and (15), we obtain
Hence, we obtain
Therefore, we have
By using (13), we immediately obtain the desired inequality (11). The condition of the equality for inequality (11) can easily be obtained by Lemma 2.3. The proof of Theorem 2.5 is completed. □
If we set , then from Theorem 2.5, we obtain the following reversed version of inequality (3).
Corollary 2.6 Let , , (), , , , . Then
If we set , , , , (), then from Theorem 2.5, we obtain
Corollary 2.7 Let , (), , , , , . Then the following inequality holds:
Remark 2.8 For , inequality (21) reduces to the famous Aczél-Vasić-Pečarić inequality (2).
As application of the above results, we establish here an integral type of the reversed version of the Aczél-Vasić-Pečarić inequality.
Theorem 3.1 Let , (), , let (), and let () be positive Riemann integrable functions on such that . Then
Proof For any positive integer n, we choose an equidistant partition of as
Since the hypothesis () implies that
there exists a positive integer N such that
By using Theorem 2.5, we obtain that for any , the following inequality holds:
In view of the hypotheses that () are positive Riemann integrable functions on , we conclude that and are also integrable on . Passing the limit as on both sides of inequality (24), we obtain inequality (22). The proof of Theorem 3.1 is completed. □
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The author would like to express his sincere thanks to the anonymous referees for their great efforts to improve this paper. This work was supported by the NNSF of China (Grant No. 61073121), and the Fundamental Research Funds for the Central Universities (No. 11ML65).
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Tian, JF. Reversed version of a generalized Aczél’s inequality and its application. J Inequal Appl 2012, 202 (2012). https://doi.org/10.1186/1029-242X-2012-202
- Aczél’s inequality
- Aczél-Vasić-Pečarić inequality
- reversed version