A New Refinement of Generalized Hölder’s Inequality and Its Application
We present a new refinement of generalized Hölder’s inequality due to Vasić and Pečarić. Moreover, the obtained result is used to improve Beckenbach-type inequality due to Wang.
If , , , and , then The sign of inequality is reversed for , (for ; we assume that ). Inequality (1) and its reversed version are called Hölder’s inequalities and are important in the study of inequalities and in the field of applied mathematics. The important inequalities have attracted interest of many mathematicians and have been improved as well as generalized in several different directions. For example, Barza et al.  presented matriceal versions of Hölder’s inequality. Nikolova and Varošanec  obtained some new refinements of the classical Hölder’s inequality by using a convex function. Tian and Hu  established a new reversed version of a generalized sharp Hölder’s inequality. For more detailed expositions, the interested reader may consult [1–13] and the references therein. Among various generalizations of (1), Vasić and Pečarić in  presented the following interesting theorem.
Theorem A. Let .(a)If are positive numbers, such that , then (b)If , and if , then (c)If , then
The main objective of this paper is to build some new refinements of inequalities (2), (3), and (4). Moreover, the obtained results will be applied to improve Beckenbach-type inequality which is due to Wang .
2. A New Refinement of Generalized Hölder’s Inequality
In this section, we first prove the following lemma, which plays a crucial role in proving our main results.
Lemma 1. Let and let .(a) If and if , then where .(b)If , and if , then (c)If , then
Proof. (a) Without loss of generality, we assume that .
Case 1 (when ). It implies that and . According to , by using inequality (2), we have which means that the desired inequality (5) holds for .
Case 2 (when ). By applying inequality (2), we obtain That is, inequality (5) is true for .
(b) If , , then , . By the same method as in Case 1, we obtain the desired inequality (6).
(c) The proof of inequality (7) is similar to the one of inequality (5), and we omit it.
The proof of Lemma 1 is completed.
Theorem 2. Let , and let be any given natural number .(a) If and if , then where .(b)If , and if , then (c)If , then
Proof. Consider the following substitution:
It is easy to see that, for any given natural number , the following inequalities hold:
Consequently, by using the substitution (13) and inequality (5), we have for , , and thus we have that is, So, we have the desired inequality (10). The proof of inequalities (11) and (12) is similar to the one of inequality (10), and we omit it. The proof of Theorem 2 is completed.
Corollary 3. Let .(a)If and if , then where .(b)If and if , then (c)If , then
In this section, we present a refinement of Beckenbach-type inequality by using Corollary 3. The classical Beckenbach inequality was proved by Beckenbach in . Since Beckenbach discovered this inequality, it has been discussed by many researchers, who either improved it using various techniques or generalized it in many different ways. The interested reader may refer to [7, 16] and references therein. In 1983, Wang  established the following Beckenbach-type inequality.
Theorem B. Let and be positive integrable functions defined on , and let . If , then, for any positive numbers , , and , the inequality holds, where . The sign of the inequality in (21) is reversed if .
Theorem 4. Let and be positive integrable functions defined on , and let . If , then, for any positive numbers , , and , the inequality holds, where . The sign of the inequality in (22) is reversed if .
Proof. After some simple calculations, we have On the other hand, putting , , in (18), from the integral form of Hölder’s inequality (1) and Corollary 3, we obtain that is, Combining inequalities (23) and (25) yields inequality (22). In a similar way, we can prove that the reversed version of inequality (22) is true. Thus, the proof of Theorem 4 is complete.
The author would like to sincerely declare his special thanks to both the anonymous referees for their helpful comments and suggestions. This work was supported by the NNSF of China (Grant no. 61073121) and the Fundamental Research Funds for the Central Universities (Grant no. 13ZD19).
D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, vol. 61 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.View at: MathSciNet
J. Tian, “Reversed version of a generalized Aczel's inequality and its application,” Journal of Inequalities and Applications, vol. 2012, article 202, 2012.View at: Google Scholar
J. Tian, “Property of a Holder-type inequality and its application,” Mathematical Inequalities and Applications, vol. 16, no. 3, pp. 831–841, 2013.View at: Google Scholar
J. Tian and X. M. Hu, “Renements of generalized Hölder's inequality,” Journal of Mathematical Inequalities. In press.View at: Google Scholar
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 2nd edition, 1952.View at: MathSciNet