Refinements of Aczél-Type Inequality and Their Applications
We present some new sharpened versions of Aczél-type inequality. Moreover, as applications, some refinements of integral type of Aczél-type inequality are given.
Let be a positive integer, and let , be real numbers such that or . Then, the famous Aczél inequality  can be stated as follows:
Aczél’s inequality plays a very important role in the theory of functional equations in non-Euclidean geometry. Due to the importance of Aczél’s inequality (1), it has received considerable attention by many authors and has motivated a large number of research papers giving it various generalizations, improvements, and applications (see [2–21] and the references therein).
In 1959, Popoviciu  first obtained an exponential extension of the Aczél inequality as follows.
Theorem B. Let , , and let , be positive numbers such that and . Then
Theorem C. Let , , , and let , be positive numbers such that and . Then
Theorem D. Let , , , , , and let . Then
Theorem E. Let , , , , , , . Then
Theorem F. Let , , , let , and let be positive Riemann integrable functions on such that . Then
Remark 1. In fact, the integral form of inequality (4) is also valid; that is, one has the following.
Theorem G. Let , , let , and let be positive Riemann integrable functions on such that . Then
2. Refinements of Aczél-Type Inequality
In order to present our main results, we need some lemmas as follows.
Lemma 3 (see ). Let .(a)If and if , then (b)If , then (c)If , , and , then
Lemma 4 (see ). Let , . Then
Lemma 5. Let , , , let , and let .
Proof. From the assumptions we have that
Case (I) (let be even). In view of by using inequality (9),
On the other hand, applying Lemma 4 and the arithmetic-geometric means inequality we obtain Applying Lemma 4 again, we get
Combining (15), (16), and (17) yields immediately inequality (13).
Case (II) (let be odd). In view of ++++++=++, by using inequality (9), we have On the other hand, applying Lemma 4 and the arithmetic-geometric means inequality we obtain Applying Lemma 4 again, we have Hence, combining (18), (19), and (20) yields immediately inequality (13).
Lemma 6. Let , , , , let , , and let .
Using the same methods as in Lemma 6, we get the following Lemma.
Lemma 7. Let , , let , and let .
Theorem 8. Let , , , , let , , , and let .
Proof. From the assumptions we find that
Thus, by using Lemma 5 with a substitution in (13), we obtain which implies
On the other hand, we get from Lemma 3 that Combining (26) and (27) yields immediately the desired inequality (23).
Theorem 9. Let , , , , , let , and let .
Then Inequality (28) is also valid for , , .
In this section, we show two applications of the inequalities newly obtained in Section 2.
Theorem 10. Let , , , , let be positive integrable functions defined on with , and let .
Proof. For any positive integer , we choose an equidistant partition of as
Since , it follows that Therefore, there exists a positive integer such that for all and .
Moreover, for any , it follows from Theorem 9 that
Noting that we get In view of the assumption that are positive Riemann integrable functions on , we find that and are also integrable on . Letting on both sides of inequality (36), we get the desired inequality (29).
Theorem 11. Let , , , , and let be positive integrable functions defined on with , and let .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the reviewers and the editors for their valuable suggestions and comments. This work was supported by the Fundamental Research Funds for the Central Universities (Grant no. 13ZD19).
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