- Open Access
Extension of Hu Ke's inequality and its applications
© Tian; licensee Springer. 2011
- Received: 30 April 2011
- Accepted: 6 October 2011
- Published: 6 October 2011
In this paper, we extend Hu Ke's inequality, which is a sharpness of Hölder's inequality. Moreover, the obtained results are used to improve Hao Z-C inequality and Beckenbach-type inequality that is due to Wang.
Mathematics Subject Classification (2000) Primary 26D15; Secondary 26D10
- integral inequality
- Hölder's inequality
- Hu Ke's inequality
- Hao Z-C inequality
- Beckenbach-type inequality
- arithmetic-geometric mean inequality
The inequality (1) is reversed for p < 1(p ≠ 0). (For p < 0, we assume that a k , b k > 0.)
The following generalization of (1) is given in :
As is well known, Hölder's inequality plays a very important role in different branches of modern mathematics such as linear algebra, classical real and complex analysis, probability and statistics, qualitative theory of differential equations and their applications. A large number of papers dealing with refinements, generalizations and applications of inequalities (1) and (2) and their series analogues in different ares of mathematics have appeared (see e.g. [2–30] and the references therein).
Among various refinements of (1), Hu in  established the following interesting theorems.
The integral form is as follows:
The purpose of this work is to give extensions of inequalities (3) and (4) and establish their corresponding reversed versions. Moreover, the obtained results will be applied to improve Hao Z-C inequality  and Beckenbach-type inequality that is due to Wang . The rest of this paper is organized as follows. In Section 2, we present extensions of (3) and (4) and establish their corresponding reversed versions. In Section 3, we apply the obtained results to improve Hao Z-C inequality and Beckenbach-type inequality that is due to Wang. Consequently, we obtain the refinement of arithmetic-geometric mean inequality. Finally, a brief summary is given in Section 4.
We begin this section with two lemmas, which will be used in the sequel.
The inequality is reversed for 0 < α < 1.
Next, we give an extension of Hu Ke's inequality, as follows.
The integral form is as follows:
Combining inequalities (11) and (13) leads to inequality (7) immediately.
Case (II). When k is odd, by the same method as in the above case (I), we have the inequality (8). The proof of Theorem 2.3 is complete. □
To illustrate the significance of the introduction of the sequence , let us sketch an example as follows.
Example 2.5. Let , j = 1, 2, ..., 2N, n = 1, 2, ..., 2N, N ≥ 2, let , and let . Then from the generalized Hölder inequality (2), we obtain . However, from Theorem 2.3, we obtain 0 ≤ 0.
Consequently, from Lemma 2.2 and the inequalities (16) and (17), we have the desired inequality (14). The proof of Corollary 2.6 is complete. □
It is clear that inequalities (7), (14) and (16) are sharper than the inequality (2).
Now, we present the following reversed versions of inequalities (7), (8), (9) and (10).
The integral form is as follows:
Combining inequalities (11) and (21) leads to inequality (18) immediately. The proof of Theorem 2.8 is complete. □
Proof. Making similar arguments as in the proof of Corollary 2.6, we have the desired inequalities (22) and (23). □
It is clear that inequalities (18) and (22) are sharper than the generalized Hölder inequality (5).
Now, we give here some direct consequences from Theorem 2.8 and Theorem 2.9. Putting m = 2 in (18) and (19), respectively, we obtain the following corollaries.
where λ1 > 0, λ j < 0 (j = 2, 3, ..., m), .
where a j > 0, λ j > 0(j = 1, 2, ..., k), p > 0 and . The above Hao Z-C inequality is refined by using Corollary 2.7 as follows:
Combining inequalities (32) and (31) yields inequality (29) immediately. The proof of Theorem 3.1 is complete. □
From Theorem 3.1, we have the following Corollary.
It is clear that inequality (33) is sharper than the inequality (27).
Now, we give a sharpness of Beckenbach-type inequality from Corollary 2.10. The famous Beckenbach inequality  has been generalized and extended in several directions; see, e.g., . In 1983, Wang  established the following Beckenbach-type inequality.
holds, where . The sign of the inequality in (34) is reversed if p > 1.
holds, where .
Combining inequalities (36) and (38) yields inequality (35). The proof of Theorem 3.3 is complete. □
The classical Hölder's inequality plays a very important role in both theory and applications. In this paper, we have presented an extension of Hu Ke's inequality, which is a sharp Hölder's inequality, and established their corresponding reversed versions. Moreover, we have improved Hao Z-C inequality and Beckenbach-type inequality by using the obtained results. Finally, we have obtained the refinement of arithmetic-geometric mean inequality. We think that our results will be useful for those areas in which inequalities (2) and (5) play a role. In the future research, we will continue to explore other applications of the obtained inequalities.
The author would like to express his sincere thanks to the anonymous referees for their making great efforts to improve this paper. This work was supported by the NNSF of China (Grant No. 61073121), and the Fundamental Research Funds for the Central Universities (No. 11ML65).
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