On a more accurate half-discrete Hilbert's inequality
© Huang and Yang; licensee Springer. 2012
Received: 8 March 2012
Accepted: 8 May 2012
Published: 8 May 2012
By using the way of weight coefficients and the idea of introducing parameters and by means of Hadamard's inequality, we give a more accurate half-discrete Hilbert's inequality with a best constant factor. We also consider its best extension with parameters, the equivalent forms, the operator expressions as well as some reverses.
2000 Mathematics Subject Classification: 26D15; 47A07.
Keywordsweight coefficient parameter equivalent form reverse Hilbert's inequality Hadamard's inequality
where the constant factor π is still the best possible. Inequalities (1)-(3) are important in analysis and its applications . There are lots of improvements, generalizations, and applications of inequalities (1-3), for more details, refer to literatures [5–18].
where the constant factor π is the best possible.
We also consider its best extension with parameters, the equivalent forms, the operator expressions as well as some reverses.
2 Some lemmas
Hence, we prove that (9) is valid.
where ϖ(x) and ω(n) are indicated by (7) and (8).
(ii) for p < 1(p ≠ 0), we have the reverses of (13) and (14).
Hence (14) is valid.
(ii) For 0 < p < 1(q < 0) or p < 0(0 < q < 1), using the reverse Hölder's inequality and in the same way, we have the reverses of (13) and (14).
Then by (26) and (27), (21) is valid.
In virtue of (30) and (31), (23) is valid.
3 Main results
where the constant factor is the best possible.
Hence (33) is valid, which is equivalent to (32).
Hence (34) is valid, which is equivalent to (32). It follows that (32), (33), and (34) are equivalent.
If there exists a positive number , such that (32) is still valid as we replace , by k, then in particular, (20) is valid ( are taken as (19)). Then we have (21). For ε → 0+ in (21), we have Hence, is the best value of (32). We conform that the constant factor in (33) [(34)] is the best possible, otherwise we can get a contradiction by (35) [(39)] that the constant factor in (32) is not the best possible.
- (ii)Define a half-discrete Hilbert's operator in the following way: For a ∈ l q , ψ , we define , satisfying
Then by (34), it follows i.e. is the bounded operator with Since the constant factor in (34) is the best possible, we have
where the constant factor is the best possible.
Then by (43), (42) is valid.
Hence (43) is valid, which is equivalent to (42).
Hence (44) is valid, which is equivalent to (42). It follows that (42), (43), and (44) are equivalent.
If there exists a positive number such that (42) is still valid as we replace by k, then in particular, (22) is valid. Hence we have (23). For ε → 0+ in (23), we obtain Hence is the best value of (42). We conform that the constant factor in (43) [(44)] is the best possible, otherwise we can get a contradiction by (45) [(46)] that the constant factor in (42) is not the best possible.
In the same way, for p < 0, we also have the following result:
Theorem 3 If the assumption of p > 1 in Theorem 1 is replaced by p < 0, then the reverses of (32), (33), and (34) are valid and equivalent. Moreover, the same constant factor is the best possible.
In particular, for , , p = q = 2 in (48), we obtain (5). Hence, inequality (32) is the best extension of (4) and (5) with parameters.
This study was supported by the Emphases Natural Science Foundation of Guangdong Institution, Higher Learning, College and University (No. 05Z026), and Guangdong Natural Science Foundation (No. 7004344).
- Weyl H: Singulare integral gleichungen mit besonderer berucksichtigung des fourierschen integral theorems. Inaugeral Dissertation, Gottingen 1908.Google Scholar
- Schur I: Bernerkungen sur Theorie der beschrankten Bilinearformen mit unendlich vielen veranderlichen. J Math 1911, 140: 1–28.MathSciNetMATHGoogle Scholar
- Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge; 1934.Google Scholar
- Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Acaremic Publishers, Boston; 1991.View ArticleGoogle Scholar
- Yang B: On Hilbert's integral inequality. J Math Anal Appl 1998, 220: 778–785. 10.1006/jmaa.1997.5877MathSciNetView ArticleMATHGoogle Scholar
- Yang B, Debnath L: On the extended Hardy-Hilbert's inequality. J Math Anal Appl 2002, 272: 187–199. 10.1016/S0022-247X(02)00151-8MathSciNetView ArticleMATHGoogle Scholar
- Jin J, Debnath L: On a Hilbert-type linear series operator and its applications. J Math Anal Appl 2010, 371: 691–704. 10.1016/j.jmaa.2010.06.002MathSciNetView ArticleMATHGoogle Scholar
- Yang B, Rassias T: On a new extension of Hilbert's inequality. Math Ineq Appl 2005, 8(4):575–582.MathSciNetMATHGoogle Scholar
- Krnić M, Pečarić J: Hilbert's inequalities and their reverses. Publ Math Debrecen 2005, 67(3–4):315–331.MathSciNetMATHGoogle Scholar
- Azar L: On some extensions of Hardy-Hilbert's inequality and applications. J Inequal Appl 2009., 2009: Article ID 546829Google Scholar
- Li Y, He B: On inequalities of Hilbert's type. Bull Aust Math Soc 2007, 76: 1–13. 10.1017/S0004972700039423View ArticleMATHGoogle Scholar
- Zhong W: The Hilbert-type integral inequality with a homogeneous kernel of lambda-degree. J Inequal Appl 2008., 2008: Article ID 917392Google Scholar
- Huang Q, Yang B: On a multiple Hilbert-type integral operator and applications. J Inequal Appl 2009., 2009: Article ID 192197Google Scholar
- Huang Q: On a multiple Hilbert's inequality with parameters. J Inequal Appl 2010., 2010: Article ID 309319Google Scholar
- Yang B: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing; 2009.Google Scholar
- Yang B: Hilbert-Type Integral Inequalities. Bentham Science Publishers Ltd., Dubai; 2009.Google Scholar
- Yang B: Discrete Hilbert-Type Inequalities. Bentham Science Publishers Ltd., Dubai; 2011.Google Scholar
- Kuang J: Applied Inequalities. Shangdong Science Technic Press, Jinan; 2010.Google Scholar
- Yang B: A mixed Hilbert-type inequality with a best constant factor. Int J Pure Appl Math 2005, 20(3):319–328.MathSciNetMATHGoogle Scholar
- Yang B: A half-discrete Hilbert's inequality. J Guangdong Univ Educ 2011, 31(3):1–8. in ChineseMATHGoogle Scholar
- Yang B: A half-discrete reverse Hilbert-type inequality with a homogeneous kernel of positive degree. J Zhanjiang Normal College 2011, 32(3):5–9. in ChineseGoogle Scholar
- Yang B: On a half-discrete Hilbert-type inequality. J Shantou Univ (Natural Science) 2011, 26(4):5–10. in ChineseGoogle Scholar
- Kuang J: Real and Functional Analysis. Higher Education Press, Beijing; 2002.Google Scholar
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