- Research Article
- Open Access

# Hardy-Hilbert-Type Inequalities with a Homogeneous Kernel in Discrete Case

- Josip Pečarić
^{1}and - Predrag Vuković
^{2}Email author

**2010**:912601

https://doi.org/10.1155/2010/912601

© J. Pečarić and P. Vuković 2010

**Received:**4 September 2009**Accepted:**16 February 2010**Published:**21 February 2010

## Abstract

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.

## Keywords

- Constant Factor
- Type Inequality
- Homogeneous Function
- Counting Measure
- Numerous Mathematician

## 1. Introduction

Hilbert and Hardy-Hilbert type inequalities (see [1]) 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. [1]):

where the constant factor is the best possible. The equivalent form of inequality (1.1) is (see Yang and Debnath [2])

where the constant factor is still the best possible.

In this paper we refer to a recent paper of Yang (see [3]). In 2005, Yang [3] gave some extension of Hilbert's inequality with two pairs of conjugate exponents and two parameters as

where the constant factor is the best possible.

Let and Define a Hilbert-type linear operator for all one has

For define the formal inner product of and as

Zhong (see [4]) proved the following theorem.

Theorem.

where the constant factor is the best possible.

Results in this paper will be based on the following general form of Hilbert's and Hardy-Hilbert's inequality proven in [5]. 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 [5] proved the following result.

Theorem.

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.

## 2. Main Results

We apply Theorem 1.2 to obtain the following theorem.

Theorem.

Proof.

Now, the result follows from Theorem 1.2.

Remark.

for arbitrary constants and (see [5]). Condition (2.8) immediately gives that nontrivial case of equality in (2.1) and (2.2) leads to divergent series.

Now, we consider some special choice of the parameters and More precisely, let the parameters and satisfy constraint

Then, the constant from Theorem 2.1 becomes

Further, the inequalities (2.1) and (2.2) take form

In the following theorem we show, in a similar way as Yang did in [6], 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 [6]).

Lemma.

Theorem.

Let , and be defined as in Theorem 2.1. If the parameters and satisfy condition then the constants and in the inequalities (2.11) and (2.12) are the best possible.

Proof.

Using symmetry of the function we have Now, from (2.17) we obtain a contradiction with assumption

Finally, equivalence of the inequalities (2.11) and (2.12) means that the constant is the best possible in the inequality (2.12). This completes the proof.

We proceed with some special homogeneous functions. Since the function is homogeneous of degree by using Theorem 2.4 we obtain the following.

Corollary.

where the constant factors and are the best possible.

Remark.

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.

Remark.

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 [4]).

Remark.

where the constants and are the best possible.

Remark.

where the constant factors and are the best possible.

Setting in the inequalities (2.21) we obtain the result from [6]. Similarly, for above choice of the parameters , and we obtain Yang's result (1.3) from Introduction.

## Authors’ Affiliations

## References

- 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.
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*Mathematical Inequalities & Applications*2005, 8(1):29–52.MathSciNetView ArticleGoogle Scholar - Yang B: On a Hilbert-type operator with a class of homogeneous kernels.
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## Copyright

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.