A New Reversed Version of a Generalized Sharp Hölder's Inequality and Its Applications

Abstract

We present a new reversed version of a generalized sharp Hölder's inequality which is due to Wu and then give a new refinement of Hölder's inequality. Moreover, the obtained result is used to improve the well-known Popoviciu-Vasić inequality. Finally, we establish the time scales version of Beckenbach-type inequality.

1. Introduction

The classical Hölder's inequality states that if 𝑎_𝑘≥0, 𝑏_𝑘≥0(𝑘=1,2,…,𝑛), 𝑝>1, and 1/𝑝+1/𝑞=1, then ^𝑛_𝑘=1𝑎_𝑘𝑏_𝑘≤^𝑛_𝑘=1𝑎^𝑝_𝑘^1/𝑝𝑛_𝑘=1𝑏^𝑞_𝑘^1/𝑞. (1) The inequality is reversed for 𝑝<1 (𝑝≠0), (for 𝑝<0, we assume that 𝑎_𝑘,𝑏_𝑘>0).

As is well known, Hölder's inequality plays an important role in different branches of modern mathematics such as classical real and complex analysis, probability and statistics, numerical analysis, and qualitative theory of differential equations and their applications. Various refinements, generalizations, and applications of inequality (1) and its series analogues in different areas of mathematics have appeared in the literature. For example, Abramovich et al. [1] presented a new generalization of Hölder's inequality and its reversed version in discrete and integral forms. Ivanković et al. [2] presented the properties of several mappings which have arisen from the Minkowski inequality and then gave some refinements of the Hölder inequality. Liu [3] obtained Hölder's inequality in fuzzy set theory and rough set theory. Nikolova and Varošanec [4] obtained some new refinements of the classical Hölder inequality by using a convex function. For detailed expositions, the interested reader may consult [1–18] and the references therein.

Among various refinements of (1), Hu in [9] established the following interesting sharpness of Hölder's inequality.

Theorem A. Let 𝑝≥𝑞>0, 1/𝑝+1/𝑞=1, let 𝐴_𝑛,𝐵_𝑛≥0, ∑_𝑛𝐴^𝑝_𝑛<∞, and ∑_𝑛𝐵^𝑞_𝑛<∞, and let 1−𝑒_𝑛+𝑒_𝑚≥0, ∑_𝑛|𝑒_𝑛|<∞. Then, _𝑛𝐴_𝑛𝐵_𝑛≤_𝑛𝐵^𝑞_𝑛^{1/𝑞−1/𝑝}×_𝑛𝐵^𝑞_𝑛𝑛𝐴^𝑝_𝑛²−_𝑛𝐵^𝑞_𝑛𝑒_𝑛𝑛𝐴^𝑝_𝑛−_𝑛𝐵^𝑞_𝑛𝑛𝐴^𝑝_𝑛𝑒_𝑛^21/2𝑝. (2)

In 2007, Wu [18] presented the generalization of Hu's result as follows.

Theorem B. Let 𝐴_𝑟≥0, 𝐵_𝑟>0(𝑟=1,2,…,𝑛), let 1−𝑒_𝑟+𝑒_𝑠≥0(𝑟,𝑠=1,2,…,𝑛), and let 𝑝≥𝑞>0, 𝜇=min{1/𝑝+1/𝑞,1}. Then, ^𝑛_𝑟=1𝐴_𝑟𝐵_𝑟≤𝑛^1−𝜇𝑛_𝑟=1𝐵^𝑞_𝑟^{1/𝑞−1/𝑝}×^𝑛_𝑟=1𝐵^𝑞_𝑟^𝑛_𝑟=1𝐴^𝑝_𝑟²−^𝑛_𝑟=1𝐵^𝑞_𝑟𝑒_𝑟^𝑛_𝑟=1𝐴^𝑝_𝑟−^𝑛_𝑟=1𝐵^𝑞_𝑟^𝑛_𝑟=1𝐴^𝑝_𝑟𝑒_𝑟^21/2𝑝. (3)

Theorem C. Let 𝑓(𝑥), 𝑔(𝑥), and 𝑒(𝑥) be integrable functions defined on [𝑎,𝑏] and 𝑓(𝑥)≥0, 𝑔(𝑥)>0, 1−𝑒(𝑥)+𝑒(𝑦)≥0 for all 𝑥,𝑦∈[𝑎,𝑏], and let 𝑝≥𝑞>0, 1/𝑝+1/𝑞≤1. Then, ^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)≤()𝑑𝑥𝑏−𝑎^{1−1/𝑝−1/𝑞𝑏}_𝑎𝑔^𝑞(𝑥)𝑑𝑥^{1/𝑞−1/𝑝}×^𝑏_𝑎𝑔^𝑞(𝑥)𝑑𝑥^𝑏_𝑎𝑓^𝑝(𝑥)𝑑𝑥²−^𝑏_𝑎𝑔^𝑞(𝑥)𝑒(𝑥)𝑑𝑥^𝑏_𝑎𝑓^𝑝(𝑥)−𝑑𝑥_𝑏𝑎𝑔^𝑞(𝑥)𝑑𝑥^𝑏_𝑎𝑓^𝑝(𝑥)𝑒(𝑥)𝑑𝑥^21/2𝑝. (4)

Recently, Tian in [13] proved the following reversed versions of inequalities (3) and (4).

Theorem D. Let 𝐴_𝑟>0, 𝐵_𝑟>0(𝑟=1,2,…,𝑛), let 1−𝑒_𝑟+𝑒_𝑠≥0(𝑟,𝑠=1,2,…,𝑛), and let 𝑞<0, 1/𝑝+1/𝑞≥0, 𝜇=max{1/𝑝+1/𝑞,1}, 𝜆=max{1/𝑞,−1}. Then, ^𝑛_𝑟=1𝐴_𝑟𝐵_𝑟≤𝑛^1−𝜇𝑛_𝑟=1𝐴^𝑝_𝑟^1/𝑝𝑛_𝑟=1𝐵^𝑞_𝑟^1/𝑞×∑1−^𝑛𝑟=1𝐴_𝑟𝐵_𝑟𝑒_𝑟∑^𝑛_𝑟=1𝐴_𝑟𝐵_𝑟−∑^𝑛_𝑟=1𝐵^𝑞_𝑟𝑒_𝑟∑^𝑛_𝑟=1𝐵^𝑞_𝑟^2𝜆/2. (5)

Theorem E. Let 𝑓(𝑥), 𝑔(𝑥), and 𝑒(𝑥) be integrable functions defined on [𝑎,𝑏] and 𝑓(𝑥)>0, 𝑔(𝑥)>0, and 1−𝑒(𝑥)+𝑒(𝑦)≥0 for all 𝑥,𝑦∈[𝑎,𝑏], and let 𝑞<0, 1/𝑝+1/𝑞≥1, 𝜆=max{−1,1/𝑞}. Then, ^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)d𝑥≥(𝑏−𝑎)^{1−1/𝑝−1/𝑞}×^𝑏_𝑎𝑓^𝑝(𝑥)d𝑥^1/𝑝𝑏_𝑎𝑔^𝑞(𝑥)d𝑥^1/𝑞×∫1−^𝑏𝑎𝑓(𝑥)𝑔(𝑥)𝑒(𝑥)d𝑥∫^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)−∫d𝑥_𝑏𝑎𝑔^𝑞(𝑥)𝑒(𝑥)d𝑥∫^𝑏_𝑎𝑔^𝑞(𝑥)d𝑥^2𝜆/2. (6)

The aim of this paper is to give new reversed versions of (3) and (4). Moreover, two applications of the obtained results are presented. The rest of this paper is organized as follows. In Section 2, we present reversed versions of (3) and (4). Moreover, we give a new refinement of Hölder's inequality. In Section 3, we apply the obtained result to improve the Popoviciu-Vasić inequality. Furthermore, we establish the time scales version of Beckenbach-type inequality.

2. A New Reversed Version of a Generalized Sharp Hölder's Inequality

In order to prove the main results, we need the following lemmas.

Lemma 1 (see, e.g., [11, page 12]). Let 𝐴_𝑘𝑗>0(𝑗=1,2,…,𝑚,𝑘=1,2,…,𝑛), ∑^𝑚_𝑗=11/𝑝_𝑗≤1. If 𝑝₁>0, 𝑝_𝑗<0(𝑗=2,3,…,𝑚), then ^𝑛𝑚_𝑘=1𝑗=1𝐴_𝑘𝑗≥^𝑚_𝑗=1^𝑛_𝑘=1𝐴^𝑝𝑗_𝑘𝑗^1/𝑝𝑗. (7)

Lemma 2 (see [19, page 12]). If 𝑥>−1, 𝛼≥1, or 𝛼<0, then (1+𝑥)^𝛼≥1+𝛼𝑥. (8) The inequality is reversed for 0<𝛼≤1.

Lemma 3 (see [7, page 27]). If 𝑥_𝑖≥0, 𝜆_𝑖>0, 𝑖=1,2,…,𝑛, and 0<𝑝≤1, then ^𝑛_𝑖=1𝜆_𝑖𝑥^𝑝_𝑖≤^𝑛_𝑖=1𝜆_𝑖^1−𝑝𝑛_𝑖=1𝜆_𝑖𝑥_𝑖^𝑝. (9) The inequality is reversed for 𝑝≥1 or 𝑝<0.

Next, we give a reversed version of inequality (3) as follows.

Theorem 4. Let 𝐴_𝑟≥0, 𝐵_𝑟>0(𝑟=1,2,…,𝑛), let 1−𝑒_𝑟+𝑒_𝑠≥0(𝑟,𝑠=1,2,…,𝑛), and let 𝑞<0, 𝑝>0, 𝜌=max{1/𝑝+1/𝑞,1}. Then, ^𝑛_𝑟=1𝐴_𝑟𝐵_𝑟≥𝑛^1−𝜌𝑛_𝑟=1𝐴^𝑝_𝑟^{1/𝑝−1/𝑞}×^𝑛_𝑟=1𝐴^𝑝_𝑟^𝑛_𝑟=1𝐵^𝑞_𝑟²−^𝑛_𝑟=1𝐴^𝑝_𝑟𝑒_𝑟^𝑛_𝑟=1𝐵^𝑞_𝑟−^𝑛_𝑟=1𝐴^𝑝_𝑟^𝑛_𝑟=1𝐵^𝑞_𝑟𝑒_𝑟^21/2𝑞. (10)

Proof. We first consider the case (I)1/𝑝+1/𝑞≤1. On one hand, performing some simple computations, we have ^𝑛_𝑟=1𝐴_𝑟𝐵_𝑟^𝑛_𝑠=1𝐴_𝑠𝐵_𝑠1−𝑒_𝑟+𝑒_𝑠=^𝑛𝑛_𝑠=1𝑟=1𝐴_𝑟𝐵_𝑟𝐴_𝑠𝐵_𝑠−^𝑛𝑛_𝑠=1𝑟=1𝐴_𝑟𝐵_𝑟𝐴_𝑠𝐵_𝑠𝑒_𝑟+^𝑛𝑛_𝑠=1𝑟=1𝐴_𝑟𝐵_𝑟𝐴_𝑠𝐵_𝑠𝑒_𝑠=^𝑛_𝑘=1𝐴_𝑘𝐵_𝑘². (11)
On the other hand, by using inequality (9), we have ^𝑛_𝑟=1𝐴_𝑟𝐵_𝑟^𝑛_𝑠=1𝐴_𝑠𝐵_𝑠1−𝑒_𝑟+𝑒_𝑠^{1/𝑝+1/𝑞}=^𝑛𝑛_𝑟=1𝑠=1𝐴_𝑟𝐵_𝑟𝐴_𝑠𝐵_𝑠1−𝑒_𝑟+𝑒_𝑠^{1/𝑝+1/𝑞}≤^𝑛𝑛_𝑟=1𝑠=1𝐴_𝑟𝐵_𝑟𝐴_𝑠𝐵_𝑠^{1−1/𝑝−1/𝑞}×^𝑛𝑛_𝑟=1𝑠=1𝐴_𝑟𝐵_𝑟𝐴_𝑠𝐵_𝑠1−𝑒_𝑟+𝑒_𝑠^{1/𝑝+1/𝑞}=^𝑛𝑛_𝑟=1𝑠=1𝐴_𝑟𝐵_𝑟A_𝑠𝐵_𝑠^{1−1/𝑝−1/𝑞}×^𝑛𝑛_𝑟=1𝑠=1𝐴_𝑟𝐵_𝑟𝐴_𝑠𝐵_𝑠−^𝑛𝑛_𝑟=1𝑠=1𝐴_𝑟𝐵_𝑟𝐴_𝑠𝐵_𝑠𝑒_𝑟+^𝑛𝑛_𝑟=1𝑠=1𝐴_𝑟𝐵_𝑟𝐴_𝑠𝐵_𝑠𝑒_𝑠^{1/𝑝+1/𝑞}=^𝑛𝑛_𝑟=1𝑠=1𝐴_𝑟𝐵_𝑟𝐴_𝑠𝐵_𝑠^{1−1/𝑝−1/𝑞}×^𝑛𝑛_𝑟=1𝑠=1𝐴_𝑟𝐵_𝑟𝐴_𝑠𝐵_𝑠^{1/𝑝+1/𝑞}=^𝑛𝑛_𝑟=1𝑠=1𝐴_𝑟𝐵_𝑟𝐴_𝑠𝐵_𝑠=^𝑛_𝑟=1𝐴_𝑟𝐵_𝑟². (12) By using inequality (7), we have ^𝑛_𝑟=1𝐴_𝑟𝐵_𝑟^𝑛_𝑠=1𝐴_𝑠𝐵_𝑠1−𝑒_𝑟+𝑒_𝑠^{1/𝑝+1/𝑞}≥^𝑛_𝑟=1𝐴_𝑟𝐵_𝑟^𝑛_𝑠=1𝐴^𝑝_𝑠1−𝑒_𝑟+𝑒_𝑠^1/𝑝×^𝑛_𝑠=1𝐵^𝑞_𝑠1−𝑒_𝑟+𝑒_𝑠^1/𝑞=^𝑛_𝑟=1^𝑛_𝑠=1𝐴^𝑝_𝑟𝐴^𝑝_𝑠1−𝑒_𝑟+𝑒_𝑠^{1/𝑝−1/𝑞}×^𝑛_𝑠=1𝐴^𝑝_𝑟𝐵^𝑞_𝑠1−𝑒_𝑟+𝑒_𝑠^1/𝑞×^𝑛_𝑠=1𝐵^𝑞_𝑟𝐴^𝑝_𝑠1−𝑒_𝑟+𝑒_𝑠^1/𝑞. (13)
Consequently, according to (1/𝑝−1/𝑞)+1/𝑞+1/𝑞≤1, by using inequality (7) on the right side of (13), we observe that ^𝑛_𝑟=1𝐴_𝑟𝐵_𝑟^𝑛_𝑠=1𝐴_𝑠𝐵_𝑠1−𝑒_𝑟+𝑒_𝑠^{1/𝑝+1/𝑞}≥^𝑛𝑛_𝑟=1𝑠=1𝐴^𝑝_𝑟𝐴^𝑝_𝑠1−𝑒_𝑟+𝑒_𝑠^{1/𝑝−1/𝑞}×^𝑛𝑛_𝑟=1𝑠=1𝐴^𝑝_𝑟𝐵^𝑞_𝑠1−𝑒_𝑟+𝑒_𝑠^1/𝑞×^𝑛𝑛_𝑟=1𝑠=1𝐵^𝑞_𝑟𝐴^𝑝_𝑠1−𝑒_𝑟+𝑒_𝑠^1/𝑞=^𝑛_𝑟=1𝐴^𝑝_𝑟^{2/𝑝−2/𝑞}×^𝑛_𝑟=1𝐴^𝑝_𝑟^𝑛_𝑠=1𝐵^𝑞_𝑠−^𝑛_𝑟=1𝐴^𝑝_𝑟𝑒_𝑟^𝑛_𝑠=1𝐵^𝑞_𝑠+^𝑛_𝑟=1𝐴^𝑝_𝑟^𝑛_𝑠=1𝐵^𝑞_𝑠𝑒_𝑠×^𝑛_𝑟=1𝐵^𝑞_𝑟^𝑛_𝑠=1𝐴^𝑝_𝑠−^𝑛_𝑟=1𝐵^𝑞_𝑟𝑒_𝑟^𝑛_𝑠=1𝐴^𝑝_𝑠+^𝑛_𝑟=1𝐵^𝑞_𝑟^𝑛_𝑠=1𝐴^𝑝_𝑠𝑒_𝑠^1/𝑞=^𝑛_𝑟=1𝐴^𝑝_𝑟^{2/𝑝−2/𝑞}×^𝑛_𝑟=1𝐴^𝑝_𝑟^𝑛_𝑟=1𝐵^𝑞_𝑟²−^𝑛_𝑟=1𝐴^𝑝_𝑟𝑒_𝑟^𝑛_𝑟=1𝐵^𝑞_𝑟−^𝑛_𝑟=1𝐴^𝑝_𝑟^𝑛_𝑟=1𝐵^𝑞_𝑟𝑒_𝑟^21/𝑞. (14) Combining inequalities (12) and (14) leads to inequality (10) immediately.
Secondly, we consider the case (II) 1/𝑝+1/𝑞≥1. Let 1/𝑝+1/𝑞=𝑡(𝑡≥1), which implies that 1/𝑝𝑡+1/𝑞𝑡=1. From Hölder's inequality and (7), we have ^𝑛_𝑟=1𝐴_𝑟𝐵_𝑟^𝑛_𝑠=1𝐴_𝑠𝐵_𝑠1−𝑒_𝑟+𝑒_𝑠=^𝑛_𝑟=1𝐴_𝑟𝐵_𝑟^𝑛_𝑠=1𝐴_𝑠𝐵_𝑠1−𝑒_𝑟+𝑒_𝑠^{1/𝑝𝑡+1/𝑞𝑡}≥^𝑛_𝑟=1𝐴_𝑟𝐵_𝑟^𝑛_𝑠=1𝐴_𝑠^𝑝𝑡1−𝑒_𝑟+𝑒_𝑠^1/𝑝𝑡×^𝑛_𝑠=1𝐵_𝑠^𝑞𝑡1−𝑒_𝑟+𝑒_𝑠^1/𝑞𝑡=^𝑛_𝑟=1^𝑛_𝑠=1𝐴_𝑟^𝑝𝑡𝐴_𝑠^𝑝𝑡1−𝑒_𝑟+𝑒_𝑠^{1/𝑝𝑡−1/𝑞𝑡}×^𝑛_𝑠=1𝐵_𝑟^𝑞𝑡𝐴_𝑠^𝑝𝑡1−𝑒_𝑟+𝑒_𝑠^1/𝑞𝑡×^𝑛_𝑠=1𝐴_𝑟^𝑝𝑡𝐵_𝑠^𝑞𝑡1−𝑒_𝑟+𝑒_𝑠^1/𝑞𝑡≥^𝑛𝑛_𝑟=1𝑠=1𝐴_𝑟^𝑝𝑡𝐴_𝑠^𝑝𝑡1−𝑒_𝑟+𝑒_𝑠^{1/𝑝𝑡−1/𝑞𝑡}×^𝑛𝑛_𝑟=1𝑠=1𝐵_𝑟^𝑞𝑡𝐴_𝑠^𝑝𝑡1−𝑒_𝑟+𝑒_𝑠^1/𝑞𝑡×^𝑛𝑛_𝑟=1𝑠=1𝐴_𝑟^𝑝𝑡𝐵_𝑠^𝑞𝑡1−𝑒_𝑟+𝑒_𝑠^1/𝑞𝑡. (15) Additionally, using Lemma 3 together with 𝑡≥1, we find ^𝑛𝑛_𝑟=1𝑠=1𝐴_𝑟^𝑝𝑡𝐴_𝑠^𝑝𝑡1−𝑒_𝑟+𝑒_𝑠^{1/𝑝𝑡−1/𝑞𝑡}×^𝑛𝑛_𝑟=1𝑠=1𝐵_𝑟^𝑞𝑡𝐴_𝑠^𝑝𝑡1−𝑒_𝑟+𝑒_𝑠^1/𝑞𝑡×^𝑛𝑛_𝑟=1𝑠=1𝐴_𝑟^𝑝𝑡𝐵_𝑠^𝑞𝑡1−𝑒_𝑟+𝑒_𝑠^1/𝑞𝑡≥^𝑛𝑛_𝑟=1𝑠=11−𝑒_𝑟+𝑒_𝑠^{(1−𝑡)(1/𝑝𝑡−1/𝑞𝑡)}×^𝑛𝑛_𝑟=1𝑠=1𝐴^𝑝_𝑟𝐴^𝑝_𝑠1−𝑒_𝑟+𝑒_𝑠^{1/𝑝−1/𝑞}×^𝑛𝑛_𝑟=1𝑠=11−𝑒_𝑟+𝑒_𝑠^{(1−𝑡)/𝑞𝑡}×^𝑛𝑛_𝑟=1𝑠=1𝐵^𝑞_𝑟𝐴^𝑝_𝑠1−𝑒_𝑟+𝑒_𝑠^1/𝑞×^𝑛𝑛_𝑟=1𝑠=11−𝑒_𝑟+𝑒_𝑠^{(1−𝑡)/𝑞𝑡}×^𝑛𝑛_𝑟=1𝑠=1𝐴^𝑝_𝑟𝐵^𝑞_𝑠1−𝑒_𝑟+𝑒_𝑠^1/𝑞=^𝑛𝑛_𝑟=1𝑠=11−𝑒_𝑟+𝑒_𝑠^1−𝑡×^𝑛𝑛_𝑟=1𝑠=1𝐴^𝑝_𝑟𝐴^𝑝_𝑠1−𝑒_𝑟+𝑒_𝑠^{1/𝑝−1/𝑞}×^𝑛𝑛_𝑟=1𝑠=1𝐵^𝑞_𝑟𝐴^𝑝_𝑠1−𝑒_𝑟+𝑒_𝑠^1/𝑞×^𝑛𝑛_𝑟=1𝑠=1𝐴^𝑝_𝑟𝐵^𝑞_𝑠1−𝑒_𝑟+𝑒_𝑠^1/𝑞=𝑛^{2−2𝑡𝑛}_𝑟=1𝐴^𝑝_𝑟^{2/𝑝−2/𝑞}×^𝑛_𝑟=1𝐵^𝑞_𝑟^𝑛_𝑠=1𝐴^𝑝_𝑠−^𝑛_𝑟=1𝐵^𝑞_𝑟𝑒_𝑟^𝑛_𝑠=1𝐴^𝑝_𝑠+^𝑛_𝑟=1𝐵^𝑞_𝑟^𝑛_𝑠=1𝐴^𝑝_𝑠𝑒_𝑠^1/𝑞×^𝑛_𝑟=1𝐴^𝑝_𝑟^𝑛_𝑠=1𝐵^𝑞_𝑠−^𝑛_𝑟=1𝐴^𝑝_𝑟𝑒_𝑟^𝑛_𝑠=1𝐵^𝑞_𝑠+^𝑛_𝑟=1𝐴^𝑝_𝑟^𝑛_𝑠=1𝐵^𝑞_𝑠𝑒_𝑠^1/𝑞=𝑛^{2(1−1/𝑝−1/𝑞)𝑛}_𝑟=1𝐴^𝑝_𝑟^{2/𝑝−2/𝑞}×^𝑛_𝑟=1𝐴^𝑝_𝑟^𝑛_𝑟=1𝐵^𝑞_𝑟²−^𝑛_𝑟=1𝐴^𝑝_𝑟𝑒_𝑟^𝑛_𝑟=1𝐵^𝑞_𝑟−^𝑛_𝑟=1𝐴^𝑝_𝑟^𝑛_𝑟=1𝐵^𝑞_𝑟𝑒_𝑟^21/𝑞. (16)
Combining inequalities (11), (15), and (16) leads to inequality (10) immediately.
The proof of Theorem 4 is completed.

From Theorem 4 and Lemma 2, we obtain the refinement of Hölder's inequality (1) as follows.

Corollary 5. Let 𝐴_𝑟>0, 𝐵_𝑟>0(𝑟=1,2,…,𝑛), let 1−𝑒_𝑟+𝑒_𝑠≥0(𝑟,𝑠=1,2,…,𝑛), and let 𝑞<0, 𝑝>0, and 𝜌=max{1/𝑝+1/𝑞,1}. Then, ^𝑛_𝑟=1𝐴_𝑟𝐵_𝑟≥𝑛^1−𝜌𝑛_𝑟=1𝐴^𝑝_𝑟^1/𝑝𝑛_𝑟=1𝐵^𝑞_𝑟^1/𝑞×11−∑2𝑞^𝑛𝑟=1𝐵^𝑞_𝑟𝑒_𝑟∑^𝑛_𝑘=1𝐵^𝑞_𝑟−∑^𝑛_𝑟=1𝐴^𝑝_𝑟𝑒_𝑟∑^𝑛_𝑟=1𝐴^𝑝_𝑟². (17)

Proof. Since ∑^𝑛_𝑟=1𝐵^𝑞_𝑟𝑒_𝑟∑^𝑛_𝑘=1𝐵^𝑞_𝑟−∑^𝑛_𝑟=1𝐴^𝑝_𝑟𝑒_𝑟∑^𝑛_𝑟=1𝐴^𝑝_𝑟<1, (18) by using Lemma 2 and Theorem 4, we obtain the assertion of the corollary. The proof of Corollary 5 is completed.

Now, we give a reversed version of inequality (4) as follows.

Theorem 6. Let 𝑓(𝑥), 𝑔(𝑥), and𝑒(𝑥) be integrable functions defined on [𝑎,𝑏] and 𝑓(𝑥), 𝑔(𝑥)>0, 1−𝑒(𝑥)+𝑒(𝑦)≥0 for all 𝑥,𝑦∈[𝑎,𝑏], and let 𝑞<0, 1/𝑝+1/𝑞≥1. Then, ^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)≥()𝑑𝑥𝑏−𝑎^{1−1/𝑝−1/𝑞𝑏}_𝑎𝑓^𝑝(𝑥)𝑑𝑥^{1/𝑝−1/𝑞}×^𝑏_𝑎𝑓^𝑝(𝑥)𝑑𝑥^𝑏_𝑎𝑔^𝑞(𝑥)𝑑𝑥²−^𝑏_𝑎𝑓^𝑝(𝑥)𝑒(𝑥)𝑑𝑥^𝑏_𝑎𝑔^𝑞(𝑥)−𝑑𝑥_𝑏𝑎𝑓^𝑝(𝑥)𝑑𝑥^𝑏_𝑎𝑔^𝑞(𝑥)𝑒(𝑥)𝑑𝑥^21/2𝑞. (19)

Proof. For any positive integer 𝑛, we choose an equidistant partition of [𝑎,𝑏] as follows: 𝑎<𝑎+𝑏−𝑎𝑛<⋯<𝑎+𝑏−𝑎𝑛𝑘<⋯<𝑎+𝑏−𝑎𝑛()𝑥𝑛−1<𝑏,_𝑘=𝑎+𝑏−𝑎𝑛𝑘,Δ𝑥_𝑘=𝑏−𝑎𝑛,𝑘=0,1,2,…,𝑛. (20)
Applying Theorem 4, we obtain the following inequality: ^𝑛_𝑘=1𝑓𝑥_𝑘𝑔𝑥_𝑘≥𝑛^{1−1/𝑝−1/𝑞𝑛}_𝑘=1𝑓^𝑝𝑥_𝑘^{1/𝑝−1/𝑞}×^𝑛_𝑘=1𝑓^𝑝𝑥_𝑘^𝑛_𝑘=1𝑔^𝑞𝑥_𝑘²−^𝑛_𝑘=1𝑓^𝑝𝑥_𝑘𝑒𝑥_𝑘^𝑛_𝑘=1𝑔^𝑞𝑥_𝑘−^𝑛_𝑘=1𝑓^𝑝𝑥_𝑘×^𝑛_𝑘=1𝑔^𝑞𝑥_𝑘𝑒𝑥_𝑘^21/2𝑞, (21) equivalently ^𝑛_𝑘=1𝑓𝑥_𝑘𝑔𝑥_𝑘𝑏−𝑎𝑛≥(𝑏−𝑎)^{1−1/𝑝−1/𝑞𝑛}_𝑘=1𝑓^𝑝𝑥_𝑘𝑏−𝑎𝑛^{1/𝑝−1/𝑞}×^𝑛_𝑘=1𝑓^𝑝𝑥_𝑘𝑏−𝑎𝑛^𝑛_𝑘=1𝑔^𝑞𝑥_𝑘𝑏−𝑎𝑛²−^𝑛_𝑘=1𝑓^𝑝𝑥_𝑘𝑒𝑥_𝑘𝑏−𝑎𝑛^𝑛_𝑘=1𝑔^𝑞𝑥_𝑘𝑏−𝑎𝑛−^𝑛_𝑘=1𝑓^𝑝𝑥_𝑘𝑏−𝑎𝑛×^𝑛_𝑘=1𝑔^𝑞𝑥_𝑘𝑒𝑥_𝑘𝑏−𝑎𝑛^21/2𝑞. (22)
In view of the hypotheses that 𝑓(𝑥), 𝑔(𝑥), and 𝑒(𝑥) are positive Riemann integrable functions on [𝑎,𝑏], we conclude that 𝑓^𝑝(𝑥), 𝑔^𝑞(𝑥), and 𝑔^𝑞(𝑥)𝑒(𝑥) are also integrable on [𝑎,𝑏]. Passing the limit as 𝑛→∞ in both sides of inequality (22), we obtain inequality (19). The proof of Theorem 6 is completed.

Remark 7. Making similar technique as in the proof of Corollary 5, from Theorem 6 we obtain ^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)≥()𝑑𝑥𝑏−𝑎^{1−1/𝑝−1/𝑞𝑏}_𝑎𝑓^𝑝(𝑥)𝑑𝑥^1/𝑝𝑏_𝑎𝑔^𝑞(𝑥)𝑑𝑥^1/𝑞×11−∫2𝑞^𝑏𝑎𝑓^𝑝(𝑥)𝑒(𝑥)𝑑𝑥∫^𝑏_𝑎𝑓^𝑝(𝑥)−∫𝑑𝑥^𝑏𝑎𝑔^𝑞(𝑥)𝑒(𝑥)𝑑𝑥∫^𝑏_𝑎𝑔^𝑞(𝑥)𝑑𝑥². (23)

3. Applications

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

Firstly, we provide an application of the obtained result to improve the Popoviciu-Vasić inequality. In 1956, Aczél [20] established the following inequality.

Theorem F. If 𝑎_𝑖, 𝑏_𝑖(𝑖=1,2,…,𝑛) are positive numbers such that 𝑎²₁−∑^𝑛_𝑖=2𝑎²_𝑖>0 or 𝑏²₁−∑^𝑛_𝑖=2𝑏²_𝑖>0, then 𝑎²₁−^𝑛_𝑖=2𝑎²_𝑖𝑏²₁−^𝑛_𝑖=2𝑏²_𝑖≤𝑎₁𝑏₁−^𝑛_𝑖=2𝑎_𝑖𝑏_𝑖². (24)

Inequality (24) is the well-known Aczél's inequality, which has many applications in the theory of functional equations in non-Euclidean geometry. Due to the importance of Aczél's inequality, this inequality has been given considerable attention by mathematicians and has motivated a large number of research papers involving different proofs, various generalizations, improvements and applications (see, e.g., [21–24] and the references therein).

One of the most important results in the references mentioned above is the exponential generalization of (24) asserted by Theorem G.

Theorem G. 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. Then, for 𝑝>1, one has 𝑎^𝑝₁−^𝑛_𝑖=2𝑎^𝑝_𝑖^1/𝑝𝑏^𝑞₁−^𝑛_𝑖=2𝑏^𝑞_𝑖^1/𝑞≤𝑎₁𝑏₁−^𝑛_𝑖=2𝑎_𝑖𝑏_𝑖. (25) If 𝑝<1(𝑝≠0), one has the reverse inequality.

Remark 8. The case 𝑝>1 of Theorem G was proved by Popoviciu [21]. The case 𝑝<1 was given in [24] by Vasić and Pečarić.

Now, we give a refinement of inequality (25) by Theorem 4 and Theorem B.

Theorem 9. Let 𝑎_𝑖,𝑏_𝑖≥0, 𝑎^𝑝₁−∑^𝑛_𝑖=2𝑎^𝑝_𝑖>0, and 𝑏^𝑞₁−∑^𝑛_𝑖=2𝑏^𝑞_𝑖>0, let 1−𝑒_𝑖+𝑒_𝑗≥0(𝑖,𝑗=1,2,…,𝑛), and let 𝜇=min{1/𝑝+1/𝑞,1}, 𝜌=max{1/𝑝+1/𝑞,1}. Then, for 𝑝≥𝑞>0, one has 𝑎^𝑝₁−^𝑛_𝑖=2𝑎^𝑝_𝑖^1/𝑝𝑏^𝑞₁−^𝑛_𝑖=2𝑏^𝑞_𝑖^1/𝑞≤𝑛^1−𝜇×𝑏₁^{1−𝑞/𝑝}×𝑏₁^2𝑞𝑎₁^2𝑝−𝑎^𝑝₁𝑏^𝑞₁𝑒₁+^𝑛_𝑖=2𝑏^𝑞_𝑖𝑒_𝑖−𝑒₁−𝑏^𝑞₁𝑎^𝑝₁𝑒₁+^𝑛_𝑖=2𝑎^𝑝_𝑖𝑒_𝑖−𝑒₁^21/2𝑞−^𝑛_𝑖=2𝑎_𝑖𝑏_𝑖. (26) If 𝑞<0, 𝑝>0, one has 𝑎^𝑝₁−^𝑛_𝑖=2𝑎^𝑝_𝑖^1/𝑝𝑏^𝑞₁−^𝑛_𝑖=2𝑏^𝑞_𝑖^1/𝑞≥𝑛^1−𝜌×𝑎₁^{1−𝑝/𝑞}×𝑎₁^2𝑝𝑏₁^2𝑞−𝑏^𝑞₁𝑎^𝑝₁𝑒₁+^𝑛_𝑖=2𝑎^𝑝_𝑖𝑒_𝑖−𝑒₁−𝑎^𝑝₁𝑏^𝑞₁𝑒₁+^𝑛_𝑖=2𝑏^𝑞_𝑖𝑒_𝑖−𝑒₁^21/2𝑞−^𝑛_𝑖=2𝑎_𝑖𝑏_𝑖. (27)

Proof. By substituting 𝐴^𝑝₁→𝑎^𝑝₁−^𝑛_𝑖=2𝑎^𝑝_𝑖,𝐵^𝑞₁→𝑏^𝑞₁−^𝑛_𝑖=2𝑏^𝑞_𝑖,𝐴_𝑖→𝑎_𝑖,𝐵_𝑖→𝑏_𝑖(𝑖=2,3,…,𝑛), (28) in (3) and (10), respectively, we get Theorem 9.

Remark 10. Let 𝑎₁≠0, 𝑏₁≠0, and let 1/𝑝+1/𝑞=1. If 𝑝≥𝑞>0, then we conclude from Theorem 9: 𝑎^𝑝₁−^𝑛_𝑖=2𝑎^𝑝_𝑖^1/𝑝𝑏^𝑞₁−^𝑛_𝑖=2𝑏^𝑞_𝑖^1/𝑞≤𝑎₁𝑏₁𝑏1−^𝑞1𝑒₁+∑^𝑛_𝑖=2𝑏^𝑞_𝑖𝑒_𝑖−𝑒₁𝑏^𝑞₁−𝑎^𝑝₁𝑒₁+∑^𝑛_𝑖=2𝑎^𝑝_𝑖𝑒_𝑖−𝑒₁𝑎^𝑝₁^21/2𝑞−^𝑛_𝑖=2𝑎_𝑖𝑏_𝑖. (29) Inequality (29) is reversed for 𝑞<0.

Next, we are to establish the time scales version of Beckenbach-type inequality which is due to Wang [25]. In 1983, Wang [25] established the following Beckenbach-type inequality.

Theorem H. Let 𝑓(𝑥), and 𝑔(𝑥) be positive integrable functions defined on [𝑠,𝑡], and let 1/𝑝+1/𝑞=1. If 0<𝑝<1, then, for any of the positive numbers: 𝑎, 𝑏, or 𝑐, the inequality 𝑎+𝑐∫^𝑡_𝑠𝑘^𝑝(𝑥)d𝑥^1/𝑝𝑏+𝑐∫^𝑡_𝑠𝑘(𝑥)𝑔(𝑥)≥d𝑥𝑎+𝑐∫^𝑡_𝑠𝑓^𝑝(𝑥)d𝑥^1/𝑝𝑏+𝑐∫^𝑡_𝑠𝑓(𝑥)𝑔(𝑥)d𝑥 (30) holds, where 𝑘(𝑥)=(𝑎𝑔(𝑥)/𝑏)^𝑞/𝑝. The sign of inequality in (30) is reversed if 𝑝>1.

In order to present the time scales version of (30), we recall the following concepts related to the notion of time scales. A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. The forward jump operator and the backward jump operator are defined by 𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡},𝜌(𝑡)∶=sup{𝑠∈𝕋∶𝑠<𝑡}, (31) (supplemented by inf∅=sup𝕋 and sup∅=inf𝕋). A point 𝑡∈𝕋 is called right scattered, right dense, left scattered, and left dense if 𝜎(𝑡)>𝑡, 𝜎(𝑡)=𝑡, 𝜌(𝑡)<𝑡, and 𝜌(𝑡)=𝑡 hold, respectively.

A function 𝑓∶𝕋→ℝ is said to be rd-continuous if it is continuous at each right dense point and if the left-sided limit exists at every left dense point. The set of all rd-continuous functions is denoted by 𝐶_rd[𝕋,ℝ].

Let 𝕋^𝑘∶=𝕋−𝑚,if𝕋hasleftscatteredpointin𝑀,𝕋,otherwise. (32) Let 𝑓 be a function defined on ℝ. Then 𝑓 is called differentiable at 𝑡∈𝕋^𝑘, with (delta) derivative 𝑓^Δ(𝑡) if given 𝜖>0; there exists a neighbourhood ℕ of 𝑡 such that 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)(𝜎(𝑡)−𝑠)≤𝜖|𝜎(𝑡)−𝑠| (33) for all 𝑠∈ℕ.

Remark 11. If   𝕋=ℝ, then 𝑓^Δ(𝑡) becomes the usual derivative; that is, 𝑓^Δ(𝑡)=𝑓^′(𝑡). If 𝕋=ℤ, then 𝑓^Δ(𝑡) reduces to the usual forward difference; that is, 𝑓^Δ(𝑡)=Δ𝑓(𝑡).

A function 𝐹∶𝕋→ℝ is called an antiderivative of 𝑓∶𝕋→ℝ provided that 𝐹^Δ=𝑓(𝑡) holds for all 𝑡∈𝕋. In this case, we define the integral of 𝑓 by ^𝑡_𝑠𝑓(𝜏)Δ𝜏=𝐹(𝑡)−𝐹(𝑠), (34) where 𝑠,𝑡∈𝕋.

Remark 12. If 𝕋=ℝ, then the time scale integral is an ordinary integral. If 𝕋=ℤ, then the time-scale integral is a sum.

For more details on time scales theory, the readers may consult [26–29] and the references therein. Now, we present the time scales version of (30) by using Corollary 5.

Theorem 13. Let 𝑓(𝑥),𝑔(𝑥), and ℎ(𝑥)∈𝐶_𝑟𝑑([𝑠,𝑡],[0,+∞)), where 𝐶_𝑟𝑑([𝑠,𝑡],[0,+∞)) denotes the set of rd-continuous functions defined by 𝐶_𝑟𝑑([𝑠,𝑡],[0,+∞))={𝛼∣𝛼∶[𝑠,𝑡]→[0,+∞) and α(t) is an rd"-" continuous function} and let 1/𝑝+1/𝑞=1. If 𝑝>1, then, for any of the positive numbers 𝑎, 𝑏, or 𝑐, the inequality 𝑎+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑘^𝑝(𝑥)Δ𝑥^1/𝑝𝑏+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑘(𝑥)𝑔(𝑥)≤∫Δ𝑥𝑎+𝑐^𝑡_𝑠ℎ(𝑥)𝑓^𝑝(𝑥)Δ𝑥^1/𝑝𝑏+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑓(𝑥)𝑔(𝑥)×1Δ𝑥1−2𝑞𝑐∫^𝑡_𝑠ℎ(𝑥)𝑓^𝑝(𝑥)Δ𝑥𝑎+𝑐∫^t_𝑠ℎ(𝑥)𝑓^𝑝(𝑥)−Δ𝑥𝑐∫^𝑡_𝑠ℎ(𝑥)𝑔^𝑞(𝑥)Δ𝑥𝑎^−𝑞/𝑝𝑏^𝑞+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑔^𝑞(𝑥)Δ𝑥² (35) holds, where 𝑘(𝑥)=(𝑎𝑔(𝑥)/𝑏)^𝑞/𝑝. The sign of inequality in (35) is reversed if   0<𝑝<1.

Proof. We only consider the case 0<𝑝<1. Noting that 1+𝑞/𝑝=𝑞, the left-hand side of (35) becomes 𝑎+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑎𝑔(𝑥)/𝑏^𝑞Δ𝑥^1/𝑝𝑏+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑎𝑔(𝑥)/𝑏^𝑞/𝑝𝑔(𝑥)=()Δ𝑥𝑎/𝑏^𝑞/𝑝𝑎(𝑏/𝑎)^𝑞+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑔^𝑞(𝑥)Δ𝑥^1/𝑝(𝑎/𝑏)^𝑞/𝑝𝑏(𝑏/𝑎)^𝑞/𝑝+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑔^𝑞(𝑥)=𝑎Δ𝑥_−𝑞/𝑝𝑏^𝑞+𝑐^𝑡_𝑠ℎ(𝑥)𝑔^𝑞(𝑥)Δ𝑥^−1/𝑞. (36) On the other hand, by using Hölder's inequality and inequality (17) for 𝑒₁=0, 𝑒₂=1, we obtain 𝑏+𝑐^𝑡_𝑠ℎ(𝑥)𝑓(𝑥)𝑔(𝑥)Δ𝑥≥𝑏+𝑐^𝑡_𝑠ℎ(𝑥)𝑓^𝑝(𝑥)Δ𝑥^1/𝑝×^𝑡_𝑠ℎ(𝑥)𝑔^𝑞(𝑥)Δ𝑥^1/𝑞=𝑎^1/𝑝𝑏𝑎^−1/𝑝+𝑐^𝑡_𝑠ℎ(𝑥)𝑓^𝑝(𝑥)Δ𝑥^1/𝑝×𝑐^𝑡_𝑠ℎ(𝑥)𝑔^𝑞(𝑥)Δ𝑥^1/𝑞≥𝑎+𝑐^𝑡_𝑠ℎ(𝑥)𝑓^𝑝(𝑥)Δ𝑥^1/𝑝×𝑎^−𝑞/𝑝𝑏^𝑞+𝑐^𝑡_𝑠ℎ(𝑥)𝑔^𝑞(𝑥)Δ𝑥^1/𝑞×11−2𝑞𝑐∫^𝑡_𝑠ℎ(𝑥)𝑓^𝑝(𝑥)Δ𝑥𝑎+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑓^𝑝(𝑥)−Δ𝑥𝑐∫^𝑡_𝑠ℎ(𝑥)𝑔^𝑞(𝑥)Δ𝑥𝑎^−𝑞/𝑝𝑏^𝑞+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑔^𝑞(𝑥)Δ𝑥². (38)
Combining inequalities (36) and (38) yields inequality (35). The proof of Theorem 13 is completed.

In (35), taking𝑐∫^𝑡_𝑠ℎ(𝑥)𝑓^𝑝(𝑥)Δ𝑥/(𝑎+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑓^𝑝(𝑥)Δ𝑥)=𝑐∫^𝑡_𝑠ℎ(𝑥)𝑔^𝑞(𝑥)Δ𝑥/(𝑎^−𝑞/𝑝𝑏^𝑞+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑔^𝑞(𝑥)Δ𝑥), from Theorem 13 we obtain the time scales version of Beckenbach-type inequality as follows.

Corollary 14. Let 𝑓(𝑥),𝑔(𝑥),and ℎ(𝑥)∈𝐶_𝑟𝑑([𝑠,𝑡],[0,+∞)), and let 1/𝑝+1/𝑞=1. If 𝑝>1, then, for any of the positive numbers 𝑎, 𝑏, or 𝑐, the inequality 𝑎+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑘^𝑝(𝑥)Δ𝑥^1/𝑝𝑏+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑘(𝑥)𝑔(𝑥)≤Δ𝑥𝑎+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑓^𝑝(𝑥)Δ𝑥^1/𝑝𝑏+𝑐∫^𝑡_𝑠ℎ(𝑥)𝑓(𝑥)𝑔(𝑥)Δ𝑥 (39) holds, where 𝑘(𝑥)=(𝑎𝑔(𝑥)/𝑏)^𝑞/𝑝. The sign of inequality in (39) is reversed if 0<𝑝<1.

Acknowledgments

This work was supported by the NNSF of China (Grant no. 61073121), the Natural Science Foundation of Hebei Province of China (Grant no. F2012402037), the Natural Science Foundation of Hebei Education Department (Grant no. Q2012046), and the Fundamental Research Funds for the Central Universities (Grants nos. 11ML65 and 13ZL). The authors would like to express their sincere thanks to the anonymous referees for their great efforts to improve this paper.