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

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.

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]):

(1.1)

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

(1.2)

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

(1.3)

where the constant factor is the best possible.

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

(1.4)

For define the formal inner product of and as

(1.5)

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

Theorem.

Suppose that and are two pairs of conjugate exponents, If then one has the equivalent inequalities as

(1.6)

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:

(1.7)

(1.8)

where

(1.9)

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.

Let with be nonnegative functions and where and are defined by (1.9). Then the following inequalities hold and are equivalent:

(1.10)

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.

Let with Let and be two nonnegative real sequences. If is homogeneous function of degree strictly decreasing in both parameters and then the following inequalities hold and are equivalent:

(2.1)

(2.2)

where and

(2.3)

Proof.

We use the inequalities (1.7), (1.8), and Theorem 1.2 with counting measure. First, we prove the inequality (2.1). Put and in the inequality (1.7). Then, we have

(2.4)

where and Since and the functions and are strictly decreasing, where we have

(2.5)

Using homogeneity of the functions and the substitution we get

(2.6)

In a similar manner we obtain

(2.7)

Now, the result follows from Theorem 1.2.

Remark.

Equality in the previous theorem is possible only if

(2.8)

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

(2.9)

Then, the constant from Theorem 2.1 becomes

(2.10)

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

(2.11)

(2.12)

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.

If is decreasing in and strictly decreasing in a subinterval of and then

(2.13)

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.

For this purpose, with , set and Now, let us suppose that there exists a smaller constant such that the inequality (2.11) is valid. Let denote the right-hand side of (2.11). Using Lemma 2.3, we have

(2.14)

Further, let denote the left-hand side of the inequality (2.11), for above choice of sequences and Applying, respectively, Lemma 2.3, Fubini's theorem, and substitution we have

(2.15)

From (2.11), (2.14), and (2.15) we get

(2.16)

By letting we obtain

(2.17)

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.

Let Suppose that the parameters satisfy condition Then the following inequalities hold and are equivalent:

(2.18)

where the constant factors and are the best possible.

Remark.

If we put in Corollary 2.5, then the inequalities (2.18) become

(2.19)

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.

Similarly as in Corollary 2.5, for the homogeneous function of degree nonnegative real sequences and the parameters we have

(2.20)

where the constants and are the best possible.

Remark.

Let , and be defined as in Theorem 2.1. Take in the inequalities (2.11) and (2.12). By using Theorem 2.4 we get equivalent inequalities for general homogeneous kernel :

(2.21)

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.