On a New Weighted Hilbert Inequality

Abstract

It is shown that a weighted Hilbert inequality for double series can be established by introducing a proper weight function. Thus, a quite sharp result of the classical Hilbert inequality for double series is obtained. And a similar result for the Hilbert integral inequality is also proved. Some applications are considered.

1. Introduction

Let and be two sequences of real numbers. It is all known that the inequality

(1.1)

is called Hilbert theorem for double series [1], where , and the constant factor in (1.1) is the best possible value. And the equality in (1.1) holds if and only if or is a zero-sequence. The corresponding integral form of (1.1) is that

(1.2)

where , and the constant factor in (1.2) is the best possible value. Recently, various improvements and extensions of (1.1) and (1.2) appear in a great deal of papers (see [2–6], etc.). The aim of the present paper is to build some new inequalities by using the weight function method and the technique of analysis, and then to study some applications of them.

First we give some lemmas.

Lemma 1.1.

Let be a positive integer. Then

(1.3)

Proof.

Let be real numbers. Then

(1.4)

where is an arbitrary constant. This result is given in the paper (see [7]). Based on this indefinite integral it is easy to deduce that the equality (1.3) holds.

Lemma 1.2.

If and , where , then

(1) and are monotonously decreasing in interval ;

• (2)
  

  (1.5)

where the weight function is defined by

(1.6)

.

Proof.

• (1) At first, notice that , hence we can write in the form of: , where and . It is obvious that the functions and are monotonously decreasing in . So, is also monotonously decreasing in . In the next place, notice that , therefore we can write in the form of: , where and . It is clear that the functions and are monotonously decreasing in . So, is also monotonously decreasing in .

• (2) Below, we need only to compute the first integral,

  

  (1.7)

By Lemma 1.1, we obtain the first integral of (1.5) at once after some simple computations and simplifications.

Similarly, the second integral of (1.5) can be gotten.

Lemma 1.3.

Let be a sequence of real numbers, and let be a real function and . If , then

(1.8)

Proof.

It is obvious that

(1.9)

We need only to show that

(1.10)

Let Then

(1.11)

Lemma 1.4.

Let be a real number, and let and be two real functions, and and , where . Then

(1.12)

.

Its proof is similar to that of Lemma 1.3. Hence, it is omitted.

2. Main Results

Theorem 2.1.

Let and be two sequences of real numbers. If and , then

(2.1)

where the weight function is defined by (1.6).

Proof.

Let be a real function and it satisfies condition First, we suppose that . By Lemma 1.3 and then applying Cauchy's inequality we have

(2.2)

where

(2.3)

It is easy to deduce that

(2.4)

Let . It is obvious that for and . Consider the function By Lemma 1.2, the function is monotonously decreasing in . Hereby, we have

(2.5)

Using (1.5), we can obtain immediately

(2.6)

Similarly, we have

(2.7)

It follows from (2.2), (2.6), and (2.7) that

(2.8)

where the weight function is defined by (1.6).

Next, consider the case for We can apply Schwarz's inequality to estimate the left-hand side of (2.1) as follows:

(2.9)

And then by using the inequality (2.8), the inequality (2.1) follows from (2.9) at once. It is obvious that the inequality (2.1) is a refinement of (1.1). Below, we give an extension of (1.2).

Theorem 2.2.

Let be a real number, and , and let and be two real functions, and and . Then

(2.10)

where the weight function is defined by

(2.11)

Specially, when , it is a refinement of (1.2).

Proof.

Let be a real function, and .

Firstly, we suppose that . Using Lemma 1.4 and then applying Cauchy's inequality we have

(2.12)

where

(2.13)

In the first place, we consider :

(2.14)

where

(2.15)

We need only to compute the weight function .

Let us select still . Then . Using (1.4), we have

(2.16)

Notice that . Hence, the function defined by (2.11) is just. So, we attain

(2.17)

Similarly, we have

(2.18)

We obtain from (2.12), (2.17), and (2.18) that

(2.19)

where the weight function is defined by (2.11).

Secondly, consider the case for . We can apply Schwarz's inequality to estimate the left-hand side of (2.10) as follows:

(2.20)

It follows from (2.19) and (2.20) that the inequality (2.10) is valid.

3. Applications

As applications, we will give some new refinements of Hardy-Littlewood's theorem and Widder's theorem below.

Let and for all . Define a sequence by

(3.1)

Hardy-Littlewood [1] proved that

(3.2)

where is the best constant that the inequality (3.2) keeps valid.

Theorem 3.1.

With the assumptions as the above-mentioned, define a sequence by . Then

(3.3)

where is defined by (1.6).

Proof.

By our assumptions, we may write in the form of:

Apply Schwarz's inequality to estimate the left-hand side of (3.3) as follows:

(3.4)

It is known from (2.8) and (3.4) that the inequality (3.3) is valid. Theorem is therefore proved.

Let and . Then

(3.5)

This is the famous Widder theorem (see [1]).

Theorem 3.2.

With the assumptions as the above-mentioned, then

(3.6)

where is defined by (2.11).

Proof.

First, we have the following relation:

(3.7)

Let . Then we have

(3.8)

where . By using (2.19), the inequality (3.6) follows from (3.8) at once.