Research Article | Open Access
A New Reversed Version of a Generalized Sharp Hölder's Inequality and Its Applications
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.
The classical Hölder's inequality states that if , , , and , then The inequality is reversed for (), (for , we assume that ).
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.  presented a new generalization of Hölder's inequality and its reversed version in discrete and integral forms. Ivanković et al.  presented the properties of several mappings which have arisen from the Minkowski inequality and then gave some refinements of the Hölder inequality. Liu  obtained Hölder's inequality in fuzzy set theory and rough set theory. Nikolova and Varošanec  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.
Theorem A. Let , , let , , and , and let , . Then,
In 2007, Wu  presented the generalization of Hu's result as follows.
Theorem B. Let , , let , and let , . Then,
Theorem C. Let , , and be integrable functions defined on and , , for all , and let , . Then,
Theorem D. Let , , let , and let , , , . Then,
Theorem E. Let , , and be integrable functions defined on and , , and for all , and let , , . Then,
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 , . If , , then
Lemma 2 (see [19, page 12]). If , , or , then The inequality is reversed for .
Lemma 3 (see [7, page 27]). If , , , and , then The inequality is reversed for or .
Next, we give a reversed version of inequality (3) as follows.
Theorem 4. Let , , let , and let , , . Then,
Proof. We first consider the case . On one hand, performing some simple computations, we have
On the other hand, by using inequality (9), we have By using inequality (7), we have
Consequently, according to , by using inequality (7) on the right side of (13), we observe that Combining inequalities (12) and (14) leads to inequality (10) immediately.
Secondly, we consider the case (II) . Let , which implies that . From Hölder's inequality and (7), we have Additionally, using Lemma 3 together with , we find
Combining inequalities (11), (15), and (16) leads to inequality (10) immediately.
The proof of Theorem 4 is completed.
Corollary 5. Let , , let , and let , , and . Then,
Now, we give a reversed version of inequality (4) as follows.
Theorem 6. Let , , and be integrable functions defined on and , , for all , and let , . Then,
Proof. For any positive integer , we choose an equidistant partition of as follows:
Applying Theorem 4, we obtain the following inequality: equivalently
In view of the hypotheses that , , and are positive Riemann integrable functions on , we conclude that , , and are also integrable on . Passing the limit as in both sides of inequality (22), we obtain inequality (19). The proof of Theorem 6 is completed.
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  established the following inequality.
Theorem F. If , are positive numbers such that or , then
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 and be real numbers such that and , and let , be positive numbers such that and . Then, for , one has If , one has the reverse inequality.
Theorem 9. Let , , and , let , and let , . Then, for , one has If , , one has
Theorem H. Let , and be positive integrable functions defined on , and let . If , then, for any of the positive numbers: , , or , the inequality holds, where . The sign of inequality in (30) is reversed if .
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 is an arbitrary nonempty closed subset of the real numbers . The forward jump operator and the backward jump operator are defined by (supplemented by and ). A point is called right scattered, right dense, left scattered, and left dense if , , , and hold, respectively.
A function 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 .
Let Let be a function defined on . Then is called differentiable at , with (delta) derivative if given ; there exists a neighbourhood of such that for all .
Remark 11. If , then becomes the usual derivative; that is, . If , then reduces to the usual forward difference; that is, .
A function is called an antiderivative of provided that holds for all . In this case, we define the integral of by where .
Remark 12. If , then the time scale integral is an ordinary integral. If , then the time-scale integral is a sum.
Theorem 13. Let , and , where denotes the set of rd-continuous functions defined by and α(t) is an rd"-" continuous function} and let . If , then, for any of the positive numbers , , or , the inequality holds, where . The sign of inequality in (35) is reversed if .
Proof. We only consider the case . Noting that , the left-hand side of (35) becomes
On the other hand, by using Hölder's inequality and inequality (17) for , , we obtain
Combining inequalities (36) and (38) yields inequality (35). The proof of Theorem 13 is completed.
Corollary 14. Let and , and let . If , then, for any of the positive numbers , , or , the inequality holds, where . The sign of inequality in (39) is reversed if .
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.
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Copyright © 2013 Jingfeng Tian and Xi-Mei Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.