A sharpened and generalized version of Aczél-Vasić-Pečarić inequality and its application
© Tian; licensee Springer. 2013
Received: 27 June 2013
Accepted: 3 October 2013
Published: 8 November 2013
In this paper, we present a sharpened and generalized version of Aczél-Vasić-Pečarić inequality. As an application, an integral type of Aczél-Vasić-Pečarić inequality is obtained.
In 1956, Aczél  established the following inequality, which is of wide application in the theory of functional equations in non-Euclidean geometry.
Later, in 1959, Popoviciu  gave a generalization of the above inequality.
In 1982, Vasić and Pečarić  presented the following reversed version of inequality (2).
Recently inequalities (2) and (3) were generalized and refined in many different ways; see, for example, [4–10] and . In , Wu established an interesting generalization of Aczél-Popoviciu inequality (2) as follows.
The main purpose of this work is to give a sharpened and generalized version of Aczél-Vasić-Pečarić inequality (3). Moreover, a new Aczél-Vasić-Pečarić type integral inequality is established.
2 A sharpened and generalized version of Aczél-Vasić-Pečarić inequality
We begin this section with some lemmas, which will be used in the sequel.
Lemma 2.1 
The inequality is reversed for . The sign of equality holds if and only if or .
Lemma 2.2 
The sign of equality holds if and only if the m sets are proportional for , or , , for .
and equality holds if and only if .
Combining inequalities (8)-(10) yields inequality (7). The condition of equality in (7) follows immediately from Lemma 2.2 and Lemma 2.1. The proof of Lemma 2.3 is completed. □
and equality holds if and only if for , or for .
In addition, the condition of equality for inequality (11) can easily be obtained by Lemma 2.1 and Lemma 2.2. The proof of Lemma 2.4 is completed. □
and equality holds if and only if and for , or for .
where the equality holds if and only if for , or for .
where the equality holds if and only if and for , or for .
Combining the above two inequalities gives the desired result.
Case (II). When , . By the same method as in the above case (I) and using Lemma 2.3 and Lemma 2.2, we get that inequality (13) is also valid. The proof of Theorem 2.5 is completed. □
Remark 2.6 If we set , , and in Theorem 2.5, then inequality (13) reduces to inequality (3).
If we set , then from Theorem 2.5 we obtain the following sharpened and generalized version of Aczél-Vasić-Pečarić inequality (3).
and equality holds if and only if and for , or for .
In particular, if we set , then from Corollary 2.7 we get the sharpened version of Aczél-Vasić-Pečarić inequality (3) as follows.
and equality holds if and only if and .
As application of the above results, we establish here an integral type of Aczél-Vasić-Pečarić inequality.
In view of the hypotheses that , are positive Riemann integrable functions on , we conclude that , and are also integrable on . Passing the limit as in both sides of inequality (22), we obtain inequality (20). The proof of Theorem 3.1 is completed. □
The author would like to express his sincere thanks to the anonymous referees for making 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. 13ZD19).
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