Extensions and Applications of Power-Type Aczél-Vasić-Pečarić’s Inequalities
In this paper, we enrich and develop power-type Aczél-Vasić-Pečarić’s inequalities. First of all, we give some new versions of theorems and corollaries about Aczél-Vasić-Pečarić’s inequalities by quoting some lemmas. Moreover, in combination with Hölder’s inequality, we give some applications of the new version of Aczél-Vasić-Pečarić’s inequality and give its proof process.
In 1956, Aczél  discovered the Aczél’s inequality.
Theorem 1. Let n be a positive integer and let , , , be positive numbers such that ; . Then
It is matter of common observation that Aczél’s inequality (1) is of great significance in the theory of functional equations in non-Euclidean geometry; meanwhile, many authors including Bellman , Hu et al., Tian [4, 5], Tian and Ha , Tian and Sun , Tian and Wu , Tian and Wang , Tian and Zhou , Wu , and Wu and Debnath [12, 13] pay more attention to this inequality and its refinements.
In 1959, Popoviciu  gave a generalization of Aczél’s inequality, as follows.
Theorem 2. Let , and , and let be positive numbers such that and . Then
Theorem 3. Let and , and let be positive integers such that . Then
Inequality (3) is known as Aczél- Vasić-Pečarić’s inequality.
Theorem 4. Let and be positive numbers such that for , and let . Then
Theorem 5. Let and , and let , , and , . Then we have where
Theorem 6. Let , , and , let , and let . Then where .
Recently, the power mean has attracted of many researcher ([16–24]), and many remarkable inequalities for the power mean including Hölder-type inequalities can be found in the literature [25–30]. The purpose of the article is to establish some distinctive versions of Aczél-Vasić-Pečarić’s inequality (3) for power mean type. As consequences, several integral inequalities of the obtained results are given.
2. Some Power Mean Types of Aczél-Vasić-Pečarić’s Inequality
Lemma 7 (see ). Let and , and let . Then
Like in , the power mean of order for positive numbers is defined by where .
Lemma 8 (see ). Let be a positive sequence, . If , then the function is increasing for fixed .
Lemma 9 (see ). Let be positive numbers, and let . Then
Putting , we get the following inequality:Let
The main conclusions of this paper are as follows.
Theorem 10. Let , , , and , and let and . Then
Proof. Simple computations lead to Since , , letCombining (15) and inequalities (10) and (8), we haveTherefore, from (15), (16), and (17), we obtain Hence, we obtain Therefore, we haveBy using (16), we immediately obtain which leads to inequality (13). The proof of Theorem 10 is accomplished.
According to Theorem 10 we can get the following corollaries.
Corollary 11. Let and , and let . Then
Corollary 12. Let and . Then
If we put in Corollary 11, the conclusions that we can draw from this are as follows.
Corollary 13. Let and , and let . Then
Let ; then from Theorem 10, we get
Corollary 14. Let , , and let . Then
Corollary 15. Let and , and let . Then
Theorem 16. Let , , and , let and , and denote . Then
Proof. For the hypotheses and , we have DenoteBy using inequality (8), simple computations lead toTherefore, from (30), (31), and (32), we obtain Hence ThusCombining (31) and inequality (35), we get which is equivalent to inequality (28). Successful proof of Theorem 16 is as follows.
Corollary 17. Let and , and let . Then
Corollary 18. Let and , and let . Then
Corollary 19. Let and , and let . Then
If we put in Theorem 16, then we have the following corollary.
Corollary 20. Let and , and let . Then
Corollary 21. Let and , and let . Then
Corollary 22. Let , , and , and let and . Then
Theorem 23. Let and , and let . Then
For the hypotheses and in Theorem 23, denote , then we have . Thus, we obtain the following corollaries.
Corollary 24. Let , , and let . Then
Corollary 25. Let and , and let . If , then
As is well known, analytic inequalities have various important applications in many branches of mathematics [33–37]. In this section, we will get some applications for this new inequality in Section 2.
Theorem 26. Let , , , , and . Thenwhere can be integrated on and .
Proof. For all positive integers , the equidistance of the partition of is selected as Noting that , we have Therefore, there is a positive integer , like that for all and .
Moreover, for any , it follows from Corollary 13 thatWe may find that are positive Riemann integrable functions on ; we know that and are also integrable on . Letting on both sides of inequality (52), we get the desired inequality (48). The proof of Theorem 26 is achieved.
Theorem 27. Let and , and let be positive Riemann integrable functions on such that . Then
Theorem 28. Let , , , , , and . Thenwhere can be integrated on and .
Corollary 29. Let , , , and . Thenwhere can be integrated on and .
Corollary 30. Let , , , , , and . Thenwhere can be integrated on and .
If we set in Corollary 30, then the following inequality holds.
Corollary 31. Let , , , , and . Thenwhere can be integrated on and .
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
This work was supported by the Fundamental Research Funds for the Central Universities (no. 2015ZD29) and the Higher School Science Research Funds of Hebei Province of China (no. Z2015137).