- Open Access
On a Hilbert-type inequality with a homogeneous kernel in ℝ2 and its equivalent form
© He; licensee Springer. 2012
- Received: 19 August 2011
- Accepted: 20 April 2012
- Published: 20 April 2012
By using the way of weight functions and the technique of real analysis, a new integral inequality with a homogeneous kernel and the best constant factor in ℝ2 is given. The equivalent form and the reverses are considered.
Mathematics Subject Classification (2000): 26D15.
- weight function
- Hölder's inequality
- equivalent form
The inequality (1.1) may be classified into several types (discrete and integral etc.), which is of great importance in analysis and its applications [1, 2]. Ever since the advent of inequality (1.1), all kinds of improvements and extensions can be seen in [3–12]. Note that the kernel of (1.1) is homogeneous of degree -1. In 2009,  reviews the negative degree homogeneous kernel of the parameterized Hilbert-type inequalities.
where the constant factor is the best possible.
Motivated by (1.2) and the technique of real analysis, we establish a new inequality in ℝ2 with a homogeneous kernel of 0-degree. Furthermore, the equivalent form and the corresponding reverse inequalities are also considered.
In what follows, α1, α2 will be real numbers such that 0 < α1 < α2 < π.
Similarly, ϖ(x) = k for x ∈ (0, ∞). Hence (2.2) is valid for x ∈ (-∞, 0) ∪ (0, ∞). □
and ϖ(x) = 2 ln (2 sin α) + (π - 2α) cot α.
Proof. It can be completed similarly by following the proof of Lemma 2.2 as long as applying the reverse Hölder's inequality , hence we omit the details. Since q < 0, thus (2.7) takes the positive inequality. □
where the constant factors and k p are both the best possible.
i.e., A|x| p f p (x) = B|y| q a.e. in (-∞,∞) × (-∞, ∞). We conform that A ≠ 0 (otherwise B = A = 0). Then a.e. in(-∞,∞), which contradicts the fact that . Hence (2.5) takes a strict inequality and the same as (2.4), thus (3.2) is valid.
Hence we obtain (3.2). Thus (3.2) and (3.1) are equivalent.
Hence k is the best value of (3.1). We conform that k p is also the best value of (3.2). Otherwise, we can get a contradiction by (3.3) that (3.1) is not the best possible. □
where the constant factors , both k p and k q are the best possible.
Hence we obtain (3.10). Thus (3.10) and (3.9) are equivalent.
By (3.11), we obtain (3.9), and it is equivalent between (3.11) and (3.9). Thus (3.9), (3.10), and (3.11) are equivalent.
By (3.14), if follows that k ≥ M for ε → 0+. Hence k is the best value of (3.9). Furthermore, the constant factors in (3.10) and (3.11) are both the best value too. Otherwise, by (3.3) or (3.13), we may get a contradiction that the constant factor in (3.9) is not the best possible. □
By Note (ii), Theorems 3.1 and 3.2, it follows that
The study was partially supported by the Emphases Natural Science Foundation of Guangdong Institution of Higher Learning, College and University (No. 05Z026).
- Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge, UK; 1934.Google Scholar
- Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and their Integrals and Derivatives. Kluwer Academic Publishers, Boston, MA; 1991.View ArticleGoogle Scholar
- Gao M: On Hilbert's inequality and its applications. J Math Anal Appl 1997, 212: 316–323.MathSciNetView ArticleGoogle Scholar
- Kuang J: On new extensions of Hilbert's integral inequality. Math Anal Appl 1999, 235: 608–614.MathSciNetView ArticleGoogle Scholar
- Pachpatte BG: On some new inequalities similar to Hilbert's inequality. J Math Anal Appl 1998, 226(3):166–179.MathSciNetView ArticleGoogle Scholar
- Sulaiman WT: Four inequalities similar to Hardy-Hilbert's integral inequality. J In-equal Pure Appl Math 2006, 7(2):8.MathSciNetGoogle Scholar
- Yang B: On the norm of an integral operator and applications. J Math Anal Appl 2006, 321: 182–192.MathSciNetView ArticleGoogle Scholar
- He B, Yang B: On a Hilbert-Type Integral Inequality with the Homogeneous Kernel of 0-Degree and the Hypergeometric Function. Math Practice Theory 2010, 40(18):203–211.MathSciNetGoogle Scholar
- Brnetić I, Pečarić J: Generalization of Hilbert's integral inequality. Math Inequal Appl 2004, 7(2):199–205.MathSciNetGoogle Scholar
- Li Y, He B: On inequalities of Hilbert's type. Bull Aust Math Soc 2007, 76(1):1–13.View ArticleGoogle Scholar
- Li Y, Wang Z, He B: Hilbert's Type Linear Operator and Some Extensions of Hilbert's Inequality. J Inequal Appl 2007, 2007: 10. Article ID 82138MathSciNetGoogle Scholar
- Zeng Z, Xie Z: On a New Hilbert-Type Integral Inequality with the Homogeneous Kernel of 0 Degree and the Integral in Whole Plane. J Inequal Appl 2010, 2010: 9. Article ID 256796MathSciNetGoogle Scholar
- Yang B: The Norm of Operator and Hilbert-Type Inequalities. Science Press, Beijing, China; 2009.Google Scholar
- Yang B: A new Hilbert-type integral inequality with some parameters. J Jilin Univ 2008, 46(6):1085–1090.MathSciNetGoogle Scholar
- Kuang J: Applied Inequalities. Shangdong Science Technic Press, Jinan, China; 2004.Google Scholar
- Kuang J: Introduction to Real Analysis. Hunan Education Press, Changsha, China; 1996.Google Scholar
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