- Research Article
- Open Access
Hardy-Hilbert-Type Inequalities with a Homogeneous Kernel in Discrete Case
© J. Pečarić and P. Vuković 2010
- Received: 4 September 2009
- Accepted: 16 February 2010
- Published: 21 February 2010
The main objective of this paper is a study of some new generalizations of Hilbert's and Hardy-Hilbert's type inequalities. We apply our general results to homogeneous functions. We shall obtain, in a similar way as Yang did in(2009), that the constant factors are the best possible when the parameters satisfy appropriate conditions.
- Constant Factor
- Type Inequality
- Homogeneous Function
- Counting Measure
- Numerous Mathematician
Hilbert and Hardy-Hilbert type inequalities (see ) are very significant weight inequalities which play an important role in many fields of mathematics. Although classical, such inequalities have attracted the interest of numerous mathematicians and have been generalized in many different ways. Also the numerous mathematicians reproved them using various techniques. Some possibilities of generalizing such inequalities are, for example, various choices of nonnegative measures, kernels, sets of integration, extension to multidimensional case, and so forth.
Similar inequalities, in operator form, appear in harmonic analysis where one investigates properties of boundedness of such operators. This is the reason why Hilbert's inequality is so popular and represents field of interest of numerous mathematicians: since Hilbert till nowadays.
We start with the following two discrete inequalities, which are the well-known Hilbert and Hardy-Hilbert type inequalities. More precisely, if such that and , then the following inequality holds (Hardy et al. ):
where the constant factor is the best possible. The equivalent form of inequality (1.1) is (see Yang and Debnath )
Zhong (see ) proved the following theorem.
Results in this paper will be based on the following general form of Hilbert's and Hardy-Hilbert's inequality proven in . All the measures are assumed to be finite on some measure space . Let with be nonnegative functions. Then the following inequalities hold and are equivalent:
It is of great importance to consider the case when the functions and defined by (1.9), are bounded. More precisely, Krnić and Pečarić in  proved the following result.
In this paper a generalization of Theorem 1.1 for a general type of homogeneous kernels is obtained. Recall that for a homogeneous function of degree , , equality is satisfied for every . Further, we define and suppose that for
In what follows, without further explanation, we assume that all series and integrals exist on the respective domains of their definitions.
We apply Theorem 1.2 to obtain the following theorem.
Now, the result follows from Theorem 1.2.
for arbitrary constants and (see ). Condition (2.8) immediately gives that nontrivial case of equality in (2.1) and (2.2) leads to divergent series.
Further, the inequalities (2.1) and (2.2) take form
In the following theorem we show, in a similar way as Yang did in , that if the parameters and satisfy condition (2.9), then one obtains the best possible constant. To prove this result we need the next lemma (see ).
where the constant factors , and are the best possible. For we obtain nonweighted case with the best possible constant Setting and in the inequalities (2.19) we obtain the inequalities (1.1) and (1.2) from Introduction.
It is easy to see that Theorem 2.4 is the generalization of Theorem 1.1. Namely, let us define , and Note that the parameters satisfy condition Then, the best possible constant from Theorem 2.4 becomes from Theorem 1.1 (see also ).
Setting in the inequalities (2.21) we obtain the result from . Similarly, for above choice of the parameters , and we obtain Yang's result (1.3) from Introduction.
- Hardy GH, Littlewood JE, Pólya G: Inequalities. 2nd edition. Cambridge University Press, Cambridge, UK; 1967.Google Scholar
- Yang B, Debnath L: On a new generalization of Hardy-Hilbert's inequality and its applications. Journal of Mathematical Analysis and Applications 1999, 233(2):484–497. 10.1006/jmaa.1999.6295MATHMathSciNetView ArticleGoogle Scholar
- Yang B: On best extensions of Hardy-Hilbert's inequality with two parameters. Journal of Inequalities in Pure and Applied Mathematics 2005, 6(3, article 81):1–15.Google Scholar
- Zhong W: A Hilbert-type linear operator with the norm and its applications. Journal of Inequalities and Applications 2009, 2009:-18.View ArticleGoogle Scholar
- Krnić M, Pečarić J: General Hilbert's and Hardy's inequalities. Mathematical Inequalities & Applications 2005, 8(1):29–52.MathSciNetView ArticleGoogle Scholar
- Yang B: On a Hilbert-type operator with a class of homogeneous kernels. Journal of Inequalities and Applications 2009, 2009:-9.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.