Refinements of Generalized Aczél's Inequality and Bellman's Inequality and Their Applications

Abstract

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.

1. Introduction

The famous Aczél’s inequality [1] states that if 𝑎_𝑖, 𝑏_𝑖(𝑖=1,2,…,𝑛) are real numbers such that 𝑎²₁−∑^𝑛_𝑖=2𝑎²_𝑖>0 and 𝑏²₁−∑^𝑛_𝑖=2𝑏²_𝑖>0, then 𝑎²₁−^𝑛_𝑖=2𝑎²_𝑖𝑏²₁−^𝑛_𝑖=2𝑏²_𝑖≤𝑎₁𝑏₁−^𝑛_𝑖=2𝑎_𝑖𝑏_𝑖². (1)

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 𝑝 and 𝑞 be real numbers such that 𝑝, 𝑞≠0, and 1/𝑝+1/𝑞=1 and let 𝑎_𝑖, 𝑏_𝑖(𝑖=1,2,…,𝑛) be positive numbers such that 𝑎^𝑝₁−∑^𝑛_𝑖=2𝑎^𝑝_𝑖>0 and 𝑏^𝑞₁−∑^𝑛_𝑖=2𝑏^𝑞_𝑖>0. If 𝑝>1, then 𝑎^𝑝₁−^𝑛_𝑖=2𝑎^𝑝_𝑖^1/𝑝𝑏^𝑞₁−^𝑛_𝑖=2𝑏^𝑞_𝑖^1/𝑞≤𝑎₁𝑏₁−^𝑛_𝑖=2𝑎_𝑖𝑏_𝑖. (2) If 𝑝<1(𝑝≠0), then the reverse inequality in (2) holds.

Remark 1. The case 𝑝>1 of Theorem A was proved by Popoviciu [8]. The case 𝑝<1 was given in [15] by Vasić and Pečarić.

Vasić and Pečarić [16] presented a further extension of inequality (1).

Theorem B. Let 𝑎_𝑟𝑗>0, 𝜆_𝑗>0, 𝑎^𝜆𝑗_1𝑗−∑^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗>0, 𝑟=1,2,…,𝑛, 𝑗=1,2,…,𝑚, and let ∑^𝑚_𝑗=1(1/𝜆_𝑗)≥1. Then ^𝑚_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗≤^𝑚_𝑗=1𝑎_1𝑗−^𝑛𝑚_𝑟=2𝑗=1𝑎_𝑟𝑗. (3)

In a recent paper [18], Wu and Debnath established an interesting generalization of Aczél’s inequality [1] as follows.

Theorem C. Let 𝑎_𝑟𝑗>0, 𝜆_𝑗>0, 𝑎^𝜆𝑗_1𝑗−∑^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗>0, 𝑟=1,2,…,𝑛, 𝑗=1,2,…,𝑚, and let 𝜌=min{∑^𝑚_𝑗=1(1/𝜆_𝑗),1}. Then ^𝑚_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗≤𝑛^𝑚1−𝜌_𝑗=1𝑎_1𝑗−^𝑛𝑚_𝑟=2𝑗=1𝑎_𝑟𝑗. (4)

In 2012, Tian [10] presented the following reversed version of inequality (4).

Theorem D. Let 𝑎_𝑟𝑗>0, 𝜆₁≠0, 𝜆_𝑗<0(𝑗=2,3,…,𝑚), 𝑎^𝜆𝑗_1𝑗−∑^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗>0, 𝑟=1,2,…,𝑛, 𝑗=1,2,…,𝑚, and let 𝜏=max{∑^𝑚_𝑗=1(1/𝜆_𝑗),1}. Then ^𝑚_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗≥𝑛^𝑚1−𝜏_𝑗=1𝑎_1𝑗−^𝑛𝑚_𝑟=2𝑗=1𝑎_𝑟𝑗. (5)

Therefore, applying the above inequality, Tian gave the reversed version of inequality (3) as follows.

Theorem E. Let 𝑎_𝑟𝑗>0, 𝜆₁≠0, 𝜆_𝑗<0(𝑗=2,3,…,𝑚), ∑^𝑚_𝑗=1(1/𝜆_𝑗)≤1, 𝑎^𝜆𝑗_1𝑗−∑^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗>0, and 𝑟=1,2,…,𝑛, 𝑗=1,2,…,𝑚. Then ^𝑚_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗≥^𝑚_𝑗=1𝑎_1𝑗−^𝑛𝑚_𝑟=2𝑗=1𝑎_𝑟𝑗. (6)

Moreover, in [10], Tian obtained the following integral form of inequality (5).

Theorem F. Let 𝜆₁>0, 𝜆_𝑗<0(𝑗=2,3,…,𝑚), ∑^𝑚_𝑗=1𝜆_𝑗=1, 𝑃_𝑗>0(𝑗=1,2,…,𝑚), and let 𝑓_𝑗(𝑥)(𝑗=1,2,…,𝑚) be positive Riemann integrable functions on [𝑎,𝑏] such that 𝑃^𝜆𝑗_𝑗−∫^𝑏_𝑎𝑓^𝜆𝑗_𝑗(𝑥)d𝑥>0. Then ^𝑚_𝑗=1𝑃^𝜆𝑗_𝑗−^𝑏_𝑎𝑓^𝜆𝑗_𝑗(𝑥)d𝑥^1/𝜆𝑗≥^𝑚_𝑗=1𝑃_𝑗−^𝑏_𝑎^𝑚_𝑗=1𝑓_𝑗(𝑥)d𝑥. (7)

Bellman inequality [20] related with Aczél’s inequality is stated as follows.

Theorem G. Let 𝑎_𝑖, 𝑏_𝑖(𝑖=1,2,…,𝑛) be positive numbers such that 𝑎^𝑝₁−∑^𝑛_𝑖=2𝑎^𝑝_𝑖>0 and 𝑏^𝑞₁−∑^𝑛_𝑖=2𝑏^𝑞_𝑖>0. If 𝑝≥1, then 𝑎^𝑝₁−^𝑛_𝑖=2𝑎^𝑝_𝑖^1/𝑝+𝑏^𝑝₁−^𝑛_𝑖=2𝑏^𝑝_𝑖^1/𝑝𝑝≤𝑎₁+𝑏₁^𝑝−^𝑛_𝑖=2𝑎_𝑖+𝑏_𝑖^𝑝. (8) If  0<𝑝<1, then the reverse inequality in (8) holds.

Remark 2. The case 𝑝>1 of Theorem G was proposed by Bellman [20]. The case 0<𝑝<1 was proved in [15] by Vasić and Pečarić.

The main purpose of this work is to give refinements of inequalities (5) and (8). As applications, some refinements of integral type of inequality (5) and (8) are given.

2. Refinements of Generalized Aczl’s Inequality and Bellman’s Inequality

Theorem 3. Let 𝑎_𝑟𝑗>0, 𝜆₁≠0, 𝜆_𝑗<0(𝑗=2,3,…,𝑚), 𝑎^𝜆𝑗_1𝑗−∑^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗>0, 𝑟=1,2,…,𝑛, 𝑗=1,2,…,𝑚, and let 𝜏=max{∑^𝑚_𝑗=1(1/𝜆_𝑗),1}. Then ^𝑚_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗≥𝑇𝑠,𝑎_𝑖𝑗≥𝑛^𝑚1−𝜏_𝑗=1𝑎_1𝑗−^𝑛𝑚_𝑟=2𝑗=1𝑎_𝑟𝑗, (9) where 𝑇𝑠,𝑎_𝑖𝑗=(𝑛+1−𝑠)^𝑚1−𝜏_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑠_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗−^𝑛𝑚_{𝑟=𝑠+1𝑗=1}𝑎_𝑟𝑗. (10)

Proof
Case (I). When ∑^𝑚_𝑗=1(1/𝜆_𝑗)<1, then 𝜏=1. On the one hand, we split the left-hand side of inequality (9) as follows: ^𝑚_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗=^𝑚_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑠_𝑟=2𝑎^𝜆𝑗_𝑟𝑗−^𝑛_𝑟=𝑠+1𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗=^𝑚_𝑗=1𝐴^𝜆𝑗_𝑗−^𝑛_𝑟=𝑠+1𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗, (11) where 𝐴_𝑗=𝑎^𝜆𝑗_1𝑗−^𝑠_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗,𝑗=1,2,…,𝑚. (12) From this hypothesis, it is immediate to obtain the inequality 𝐴^𝜆𝑗_𝑗≥^𝑛_𝑟=𝑠+1𝑎^𝜆𝑗_𝑟𝑗. (13) Thus, by using inequality (6), we have ^𝑚_𝑗=1𝐴^𝜆𝑗_𝑗−^𝑛_𝑟=𝑠+1𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗≥^𝑚_𝑗=1𝐴_𝑗−^𝑛𝑚_{𝑟=𝑠+1𝑗=1}𝑎_𝑟𝑗. (14)
On the other hand, by using inequality (6) again, we obtain the inequality ^𝑚_𝑗=1𝐴_𝑗=^𝑚_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑠_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗≥^𝑚_𝑗=1𝑎_1𝑗−^𝑠𝑚_𝑟=2𝑗=1𝑎_𝑟𝑗. (15) Combining inequalities (11), (14), and (15) we can get inequality (9).
Case (II). When ∑^𝑚_𝑗=1(1/𝜆_𝑗)≥1, using Case (I) with ∑^𝑚+1_𝑗=1(1/𝜆_𝑗)=1, we have ^𝑚+1_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗≥^𝑚+1_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑠_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗−^𝑛_𝑟=𝑠+1^𝑚+1_𝑗=1𝑎_𝑟𝑗, (16) where 𝑎^𝜆𝑗_1𝑗−∑^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗>0, 𝑎_𝑟𝑗>0, 𝜆₁≠0, 𝜆_𝑗<0(𝑗=2,3,…,𝑚+1).
Hence, taking 𝑎^𝜆𝑚+1_1(𝑚+1)=1, 𝑎^𝜆𝑚+1_2(𝑚+1)=𝑎^𝜆𝑚+1_3(𝑚+1)=⋯=𝑎^𝜆𝑚+1_𝑛(𝑚+1)=1/𝑛 into (16), we obtain 1𝑛^{1−1/𝜆1−1/𝜆2−⋯−1/𝜆𝑚𝑚}_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗≥𝑛+1−𝑠𝑛^{1−1/𝜆1−1/𝜆2−⋯−1/𝜆𝑚}×^𝑚_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑠_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗−1𝑛^{1−1/𝜆1−1/𝜆2−⋯−1/𝜆𝑚𝑛𝑚}_{𝑟=𝑠+1𝑗=1}𝑎_𝑟𝑗, (17) that is, ^𝑚_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗≥(𝑛+1−𝑠)^{1−1/𝜆1−1/𝜆2−⋯−1/𝜆𝑚}×^𝑚_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑠_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗−^𝑛𝑚_{𝑟=𝑠+1𝑗=1}𝑎_𝑟𝑗. (18) Therefore, repeating the foregoing arguments, we get (𝑛+1−𝑠)^{1−1/𝜆1−1/𝜆2−⋯−1/𝜆𝑚𝑚}_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑠_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗−^𝑛𝑚_{𝑟=𝑠+1𝑗=1}𝑎_𝑟𝑗≥𝑛^𝑚1−𝜏_𝑗=1𝑎_1𝑗−^𝑛𝑚_𝑟=2𝑗=1𝑎_𝑟𝑗. (19) Combining inequalities (18) and (19) leads to inequality (9) immediately. The proof of Theorem 3 is completed.

If we set ∑^𝑚_𝑗=1(1/𝜆_𝑗)≤1, then from Theorem 3, we obtain the following refinement of inequality (6).

Corollary 4. Let 𝑎_𝑟𝑗>0, 𝜆₁≠0, 𝜆_𝑗<0(𝑗=2,3,…,𝑚), ∑^𝑚_𝑗=1(1/𝜆_𝑗)≤1, 𝑎^𝜆𝑗_1𝑗−∑^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗>0, 𝑟=1,2,…,𝑛, and 𝑗=1,2,…,𝑚. Then ^𝑚_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗≥^𝑚_𝑗=1𝑎^𝜆𝑗_1𝑗−^𝑠_𝑟=2𝑎^𝜆𝑗_𝑟𝑗^1/𝜆𝑗−^𝑛𝑚_{𝑟=𝑠+1𝑗=1}𝑎_𝑟𝑗≥^𝑚_𝑗=1𝑎_1𝑗−^𝑛𝑚_𝑟=2𝑗=1𝑎_𝑟𝑗. (20)

Putting 𝑚=2, 𝜆₁=𝑝≠0, 𝜆₂=𝑞<0, 𝑎_𝑟1=𝑎_𝑟, and 𝑎_𝑟2=𝑏_𝑟(𝑟=1,2,…,𝑛) in Theorem 3, we obtain the refinement and generalization of Theorem A for 𝑝<1.

Corollary 5. Let 𝑎_𝑟>0, 𝑏_𝑟>0(𝑟=1,2,…,𝑛), 𝑎^𝑝₁−∑^𝑛_𝑟=2𝑎^𝑝_𝑟>0, 𝑏^𝑞₁−∑^𝑛_𝑟=2𝑏^𝑞_𝑟>0, 𝑝≠0, 𝑞<0, and 𝜌=max{1/𝑝+1/𝑞,1}. Then, the following inequality holds: 𝑎^𝑝₁−^𝑛_𝑟=2𝑎^𝑝_𝑟^1/𝑝𝑏^𝑞₁−^𝑛_𝑟=2𝑏^𝑞_𝑟^1/𝑞≥(𝑛+1−𝑠)^1−𝜌𝑎^𝑝₁−^𝑠_𝑟=2𝑎^𝑝_𝑟^1/𝑝×𝑏^𝑞₁−^𝑠_𝑟=2𝑏^𝑞_𝑟^1/𝑞−^𝑛_𝑟=𝑠+1𝑎_𝑟𝑏_𝑟≥𝑛^1−𝜌𝑎₁𝑏₁−^𝑛_𝑟=2𝑎_𝑟𝑏_𝑟. (21)

Based on the mathematical induction, it is easy to see that the following generalized Bellman’s inequality is true.

Theorem 6. Let 𝑎_𝑟𝑗>0, 𝑎^𝜆𝑗_1𝑗−∑^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗>0, 𝑟=1,2,…,𝑛, 𝑗=1,2,…,𝑚, and let 0<𝑝<1. Then ^𝑚_𝑗=1𝑎^𝑝_1𝑗−^𝑛_𝑟=2𝑎^𝑝_𝑟𝑗^1/𝑝𝑝≥^𝑚_𝑗=1𝑎_1𝑗^𝑝−^𝑛_𝑟=2^𝑚_𝑗=1𝑎_𝑟𝑗^𝑝. (22)

Next, we give a refinement of generalized Bellman’s inequality (22) as follows.

Theorem 7. Let 𝑎_𝑟𝑗>0, 𝑎^𝜆𝑗_1𝑗−∑^𝑛_𝑟=2𝑎^𝜆𝑗_𝑟𝑗>0, 𝑟=1,2,…,𝑛, 𝑗=1,2,…,𝑚, and let 0<𝑝<1. Then ^𝑚_𝑗=1𝑎^𝑝_1𝑗−^𝑛_𝑟=2𝑎^𝑝_𝑟𝑗^1/𝑝𝑝≥^𝑚_𝑗=1𝑎^𝑝_1𝑗−^𝑠_𝑟=2𝑎^𝑝_𝑟𝑗^1/𝑝𝑝−^𝑛_𝑟=𝑠+1^𝑚_𝑗=1𝑎_𝑟𝑗^𝑝≥^𝑚_𝑗=1𝑎_1𝑗^𝑝−^𝑛_𝑟=2^𝑚_𝑗=1𝑎_𝑟𝑗^𝑝. (23)

Proof. The proof of Theorem 7 is similar to the one of Theorem 3. Applying generalized Bellman’s inequality (22) twice, we can deduce the inequality (23).

3. Application

In this section, we show two applications of the inequalities newly obtained in Section 2.

Firstly, we give an improvement of inequality (7) by using Theorem 3.

Theorem 8. Let 𝐵_𝑗>0(𝑗=1,2,…,𝑚), let 𝜆₁>0, 𝜆_𝑗<0(𝑗=2,3,…,𝑚), ∑^𝑚_𝑗=1𝜆_𝑗=1, and let 𝑓_𝑗(𝑥)(𝑗=1,2,…,𝑚) be positive integrable functions defined on [𝑎,𝑏] with 𝐵^𝜆𝑗_𝑗−∫^𝑏_𝑎𝑓^𝜆𝑗_𝑗(𝑥)d𝑥>0. Then, for any 𝑡∈[𝑎,𝑏), one has ^𝑚_𝑗=1𝐵^𝜆𝑗_𝑗−^𝑏_𝑎𝑓^𝜆𝑗_𝑗(𝑥)d𝑥^1/𝜆𝑗≥^𝑚_𝑗=1𝐵^𝜆𝑗_𝑗−^𝑡_𝑎𝑓^𝜆𝑗_𝑗(𝑥)d𝑥^1/𝜆𝑗−^𝑏_𝑡^𝑚_𝑗=1𝑓_𝑗(𝑥)≥d𝑥_𝑚𝑗=1𝐵_𝑗−^𝑏_𝑎^𝑚_𝑗=1𝑓_𝑗(𝑥)d𝑥. (24)

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 𝑛 and 𝑙, we choose an equidistant partition of [𝑎,𝑡] and [𝑡,𝑏], respectively, as 𝑎<𝑎+𝑐−𝑎𝑛<⋯<𝑎+𝑐−𝑎𝑛𝑘<⋯<𝑎+𝑐−𝑎𝑛(𝑛−1)<𝑐,𝑐<𝑐+𝑏−𝑐𝑙<⋯<𝑐+𝑏−𝑐𝑙𝑖<⋯<𝑐+𝑏−𝑐𝑙()𝑥𝑙−1<𝑏,_𝑘=𝑎+𝑐−𝑎𝑛𝑘,Δ𝑥_𝑘=𝑐−𝑎𝑛𝑥,𝑘=1,2,…,𝑛,_𝑖=𝑐+𝑏−𝑐𝑙𝑖,Δ𝑥_𝑖=𝑏−𝑐𝑙,𝑖=1,2,…,𝑛. (25) Noting that 𝐵^𝜆𝑗_𝑗−∫^𝑏_𝑎𝑓^𝜆𝑗_𝑗(𝑥)d𝑥=𝐵^𝜆𝑗_𝑗−(∫^𝑐_𝑎𝑓^𝜆𝑗_𝑗(𝑥)d𝑥+∫^𝑏_𝑐𝑓^𝜆𝑗_𝑗(𝑥)d𝑥)>0(𝑗=1,2,…,𝑚), we have 𝐵^𝜆𝑗_𝑗−lim^𝑛_{𝑛→∞𝑘=1}𝑓^𝜆𝑗_𝑗𝑘()𝑎+𝑐−𝑎𝑛𝑐−𝑎𝑛+lim^𝑙_{𝑙→∞𝑖=1}𝑓^𝜆𝑗_𝑗𝑖()𝑐+𝑏−𝑐𝑙𝑏−𝑐𝑙.>0,𝑗=1,2,…,𝑚 (26) Consequently, there exists a positive integer 𝑁, such that 𝐵^𝜆𝑗_𝑗−^𝑛_𝑘=1𝑓^𝜆𝑗_𝑗𝑘()𝑎+𝑐−𝑎𝑛𝑐−𝑎𝑛+^𝑙_𝑖=1𝑓^𝜆𝑗_𝑗𝑖()𝑐+𝑏−𝑐𝑙𝑏−𝑐𝑙>0 (27) for all 𝑛,𝑙>𝑁 and 𝑗=1,2,…,𝑚.
By using Theorem 3, for any 𝑛,𝑙>𝑁, the following inequality holds: ^𝑚_𝑗=1𝐵^𝜆𝑗_𝑗−^𝑛_𝑘=1𝑓^𝜆𝑗_𝑗𝑘()𝑎+𝑐−𝑎𝑛𝑐−𝑎𝑛+^𝑙_𝑖=1𝑓^𝜆𝑗_𝑗𝑖()𝑐+𝑏−𝑐𝑙𝑏−𝑐𝑙^1/𝜆𝑗≥^𝑚_𝑗=1𝐵^𝜆𝑗_𝑗−^𝑛_𝑘=1𝑓^𝜆𝑗_𝑗𝑘()𝑎+𝑐−𝑎𝑛𝑐−𝑎𝑛^1/𝜆𝑗−^𝑙_𝑖=1^𝑚_𝑗=1𝑓_𝑗𝑖()𝑐+𝑏−𝑐𝑙𝑏−𝑐𝑙^{∑𝑚𝑗=1(1/𝜆𝑗)}. (28)
Since ^𝑚_𝑗=11𝜆_𝑗=1, (29)
we have ^𝑚_𝑗=1𝐵^𝜆𝑗_𝑗−^𝑛_𝑘=1𝑓^𝜆𝑗_𝑗𝑘()𝑎+𝑐−𝑎𝑛𝑐−𝑎𝑛+^𝑙_𝑖=1𝑓^𝜆𝑗_𝑗𝑖()𝑐+𝑏−𝑐𝑙𝑏−𝑐𝑙^1/𝜆𝑗≥^𝑚_𝑗=1𝐵^𝜆𝑗_𝑗−^𝑛_𝑘=1𝑓^𝜆𝑗_𝑗𝑘()𝑎+𝑐−𝑎𝑛𝑐−𝑎𝑛^1/𝜆𝑗−^𝑙_𝑖=1^𝑚_𝑗=1𝑓_𝑗𝑖()𝑐+𝑏−𝑐𝑙𝑏−𝑐𝑙. (30)
Noting that 𝑓_𝑗(𝑥)(𝑗=1,2,…,𝑚) are positive Riemann integrable functions on [𝑎,𝑏], we know that ∏^𝑚_𝑗=1𝑓_𝑗(𝑥) and 𝑓^𝜆𝑗_𝑗(𝑥) are also integrable on [𝑎,𝑏]. Letting 𝑛→∞ on both sides of inequality (30), we get the left side of inequality (24). The proof of Theorem 8 is completed.

We give here a direct consequence from Theorem 8. Putting 𝑚=2, 𝜆₁=𝑝, 𝜆₂=𝑞, 𝐵₁=𝑎₁, 𝐵₂=𝑏₁, 𝑓₁=𝑓, and 𝑓₂=𝑔 in (24), we obtain a special important case as follows.

Corollary 9. Let 𝑝 and 𝑞 be real numbers such that 𝑝>0, 𝑞<0, and (1/𝑝)+(1/𝑞)=1, let 𝑎₁,𝑏₁>0, and let 𝑓,𝑔 be positive integrable functions defined on [𝑎,𝑏] with 𝑎^𝑝₁−∫^𝑏_𝑎𝑓^𝑝(𝑥)d𝑥>0 and 𝑏^𝑞₁−∫^𝑏_𝑎𝑔^𝑞(𝑥)d𝑥>0. Then, for any 𝑡∈[𝑎,𝑏), one has 𝑎^𝑝₁−^𝑏_𝑎𝑓^𝑝(𝑥)d𝑥^1/𝑝𝑏^𝑞₁−^𝑏_𝑎𝑔^𝑞(𝑥)d𝑥^1/𝑞≥𝑎^𝑝₁−^𝑐_𝑎𝑓^𝑝(𝑥)d𝑥^1/𝑝𝑏^𝑞₁−^𝑐_𝑎𝑔^𝑞(𝑥)d𝑥^1/𝑞≥𝑎₁𝑏₁−^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)d𝑥. (31)

Finally, we present a refinement of integral type of generalized Bellman’s inequality.

Theorem 10. Let 𝐵_𝑗>0(𝑗=1,2,…,𝑚), let 0<𝑝<1, and let 𝑓_𝑗(𝑥)(𝑗=1,2,…,𝑚) be positive integrable functions defined on [𝑎,𝑏] with 𝐵^𝜆𝑗_𝑗−∫^𝑏_𝑎𝑓^𝜆𝑗_𝑗(𝑥)d𝑥>0. Then, for any 𝑡∈[𝑎,𝑏), one has ^𝑚_𝑗=1𝐵^𝑝_𝑗−^𝑏_𝑎𝑓^𝑝_𝑗(𝑥)d𝑥^1/𝑝𝑝≥^𝑚_𝑗=1𝐵^𝑝_𝑗−^𝑡_𝑎𝑓^𝑝_𝑗(𝑥)d𝑥^1/𝑝𝑝−^𝑏_𝑡^𝑚_𝑗=1𝑓_𝑗(𝑥)^𝑝≥d𝑥_𝑚𝑗=1𝐵_𝑗^𝑝−^𝑏_𝑎^𝑚_𝑗=1𝑓_𝑗(𝑥)^𝑝d𝑥. (32)

Proof. The proof of Theorem 10 is similar to the proof of Theorem 8.

A special case to the last theorem is as follows.

Corollary 11. Let 0<𝑝<1, let 𝑎₁,𝑏₁>0, and let 𝑓,𝑔 be positive integrable functions defined on [𝑎,𝑏] with 𝑎^𝑝₁−∫^𝑏_𝑎𝑓^𝑝(𝑥)d𝑥>0 and 𝑏^𝑞₁−∫^𝑏_𝑎𝑔^𝑞(𝑥)d𝑥>0. Then, for any 𝑡∈[𝑎,𝑏), one has 𝑎^𝑝₁−^𝑏_𝑎𝑓^𝑝(𝑥)d𝑥^1/𝑝+𝑏^𝑝₁−^𝑏_𝑎𝑔^𝑝(𝑥)d𝑥^1/𝑝𝑝≥𝑎^𝑝₁−^𝑡_𝑎𝑓^𝑝(𝑥)d𝑥^1/𝑝+𝑏^𝑝₁−^𝑡_𝑎𝑔^𝑝(𝑥)d𝑥^1/𝑝𝑝−^𝑏_𝑡𝑓(𝑥)+𝑔(𝑥)^𝑝≥𝑎d𝑥₁+𝑏₁^𝑝−^𝑏_𝑎𝑓(𝑥)+𝑔(𝑥)^𝑝d𝑥. (33)

Acknowledgments

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).