Refinements of Generalized Aczél's Inequality and Bellman's Inequality and Their Applications
We give some refinements of generalized Aczél's inequality and Bellman's inequality proposed by Tian. As applications, some refinements of integral type of generalized Aczél's inequality and Bellman's inequality are given.
The famous Aczél’s inequality  states that if , are real numbers such that and , then
It is well known that Aczél’s inequality plays an important role in the theory of functional equations in non-Euclidean geometry. In recent years, various attempts have been made by many authors to improve and generalize the Aczél’s inequality (see [2–19] and references therein). We state here some improvements of Aczél’s inequality.
One of the most important results in the references mentioned above is an exponential extension of (1), which is stated in the following theorem.
Theorem A. Let and be real numbers such that , , and and let , be positive numbers such that and . If , then If , then the reverse inequality in (2) holds.
Theorem B. Let , , , , , and let . Then
Theorem C. Let , , , , , and let . Then
Theorem D. Let , , , , , , and let . Then
Therefore, applying the above inequality, Tian gave the reversed version of inequality (3) as follows.
Theorem E. Let , , , , , and , . Then
Theorem F. Let , , , , and let be positive Riemann integrable functions on such that . Then
Bellman inequality  related with Aczél’s inequality is stated as follows.
Theorem G. Let , be positive numbers such that and . If , then If , then the reverse inequality in (8) holds.
2. Refinements of Generalized Aczl’s Inequality and Bellman’s Inequality
Theorem 3. Let , , , , , , and let . Then where
Case (I). When , then . On the one hand, we split the left-hand side of inequality (9) as follows: where From this hypothesis, it is immediate to obtain the inequality Thus, by using inequality (6), we have
On the other hand, by using inequality (6) again, we obtain the inequality Combining inequalities (11), (14), and (15) we can get inequality (9).
Case (II). When , using Case (I) with , we have where , , , .
Hence, taking , into (16), we obtain that is, Therefore, repeating the foregoing arguments, we get Combining inequalities (18) and (19) leads to inequality (9) immediately. The proof of Theorem 3 is completed.
Corollary 4. Let , , , , , , and . Then
Putting , , , , and in Theorem 3, we obtain the refinement and generalization of Theorem A for .
Corollary 5. Let , , , , , , and . Then, the following inequality holds:
Based on the mathematical induction, it is easy to see that the following generalized Bellman’s inequality is true.
Theorem 6. Let , , , , and let . Then
Next, we give a refinement of generalized Bellman’s inequality (22) as follows.
Theorem 7. Let , , , , and let . Then
In this section, we show two applications of the inequalities newly obtained in Section 2.
Theorem 8. Let , let , , , and let be positive integrable functions defined on with . Then, for any , one has
Proof. We need to prove only the left side of inequality (24). The proof of the right side of inequality (24) is similar. For any positive integers and , we choose an equidistant partition of and , respectively, as
Noting that , we have
Consequently, there exists a positive integer , such that
for all and .
By using Theorem 3, for any , the following inequality holds:
Noting that are positive Riemann integrable functions on , we know that and are also integrable on . Letting on both sides of inequality (30), we get the left side of inequality (24). The proof of Theorem 8 is completed.
Corollary 9. Let and be real numbers such that , , and , let , and let be positive integrable functions defined on with and . Then, for any , one has
Finally, we present a refinement of integral type of generalized Bellman’s inequality.
Theorem 10. Let , let , and let be positive integrable functions defined on with . Then, for any , one has
A special case to the last theorem is as follows.
Corollary 11. Let , let , and let be positive integrable functions defined on with and . Then, for any , one has
The authors would like to express hearty thanks to the anonymous referees for their great efforts to improve this paper. This work was supported by the NNSF of China (no. 61073121), the Natural Science Foundation of Hebei Province of China (no. F2012402037), the Natural Science Foundation of Hebei Education Department (no. Q2012046), and the Fundamental Research Funds for the Central Universities (no. 11ML65).
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