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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 165089, 12 pages
http://dx.doi.org/10.1155/2008/165089
Research Article

On Hilbert's Inequality for Double Series and Its Applications

Department of Mathematics and Computer Science, Normal College of Jishou University, Jishou, Hunan 416000, China

Received 2 September 2007; Accepted 26 March 2008

Academic Editor: Feng Qi

Copyright © 2008 Zhou Yu and Gao Mingzhe. 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.

Abstract

This study shows that a refinement of the Hilbert inequality for double series can be established by introducing a real function and a parameter . In particular, some sharp results of the classical Hilbert inequality are obtained by means of a sharpening of the Cauchy inequality. As applications, some refinements of both the Fejer-Riesz inequality and Hardy inequality in function are given.

1. Introduction

Let and be two sequences of complex numbers. If , then where the constant factor is the best possible. It is all known that the inequalities (1.1) and (1.2) are called the Hilbert theorem for double series. The two forms (1.1) and (1.2) on the Hilbert inequality were combined into one similar form in some papers (e.g., [1, 2], etc.), that is, Recently, the various extensions on (1.1) appeared in some papers (e.g., [3, 4], etc.). They focalize on changing the denominator of the function of the left-hand side of (1.1). Such as the denominator is replaced by in [3], and the denominator is replaced by in [4]. Some new results in these papers were yielded. The inequality (1.3) is obviously a significant refinement of (1.1) and (1.2). However, both extensions and refinements on (1.2) and (1.3) do not commonly appear in previous papers. The main purpose of the present paper is to establish both an extension and a significant refinement on (1.3). Explicitly, let be a real function and let . If the denominator of the first term of the left-hand side of (1.3) is replaced by , and the denominator of the second term of the left-hand side of (1.3) is replaced by , then a new inequality established is significant in theory and applications; and as applications, we will give both extensions and refinements on Fejer-Riesz’s inequality and Hardy’s inequality. For convenience, we introduce some notations and functions as follows: In particular, when , the notations and are denoted, respectively, by and . Throughout this paper, we will frequently use these notations, and we stipulate that denotes integer set and that , where , is an integer or .

2. Lemmas

In order to prove our assertions, we need the following lemmas.

Lemma 2.1. If both and are absolute convergent, then (i) is absolute convergent,(ii) and are convergent.

The proof of it has been given in the paper [2]. Hence, it is omitted here.

Lemma 2.2. Let , and let be a variable unit-vector. Then, In particular, when is orthogonal to , we have and the equality in (2.2) holds if and only if and are linearly dependent.

The proof of these results has been given in [5, 6].

Define a function by

Lemma 2.3. Let and be two sequences of complex numbers. If is an integer or , then

Proof. When is an integer, it is clear that . So we consider only the case for . It is easy to deduce that When , it follows from (2.3) that for any . Hence, we have .

Lemma 2.4. Let If is analytic in the unit-disc , then

Proof. Thereby, the relation (2.6) holds.

3. Theorems and Their Corollaries

In order to prove our assertions, we need also to introduce the following functions:

Theorem 3.1. Let be a function defined by (2.3), let and be two nonzero sequences of complex numbers, and let both and be absolute convergent. Then,

(i) if is an integer, then where

(ii) if , then where , is defined by (3.1). In particular, when , we have .

Proof. Define two functions by Since both and are absolute convergent by Lemma 2.1, the double series is absolute convergent. Accordingly, is uniformly convergent in the interval . Thereby, the interchange in order of summation and integration can be made. In what follows, we stipulate that the interchanges in order of summation and integration are justified. It is easy to deduce that where is a function defined by (2.3). By Lemma 2.2, we have where , is a variable unit-vector, it can be properly chosen in accordance with our requirement.

(i) When is an integer, it is known from (2.3) that .

We select it is easy to deduce that and Since the series is absolute convergent, it is justified that the complex number is replaced by in (3.8). Hence, we have Similarly

We therefore obtain that Hence, the inequality (3.7) can be reduced to

Notice that and . If we still select the unit-vector , then, interchanging and in (3.12), we have where is defined by

Adding (3.12) and (3.13), then the inequality (3.2) follows after simplifications.

(ii) When , we firstly consider in (3.7). We still select unit-vector . It is easy to deduce that Since the series and are absolute convergent, it is justified that the complex numbers and are replaced, respectively, by and in the above relations. Let By using (3.1), we find , . Let . We obtain from (3.7)

Notice that , and . If we still select the unit-vector , then, interchanging and in (3.16), we have where . Let . Adding (3.16) and (3.17), the inequality (3.4) can be gotten after simplifications. In particular, when , by Lemma 2.3, we have . The proof of Theorem 3.1 is completed.

Corollary 3.2 . Let be absolute convergent. Then, (i)if is an integer, then where (ii)if , then where is defined by (3.1). In particular, when , according to (3.2), one obtains a refinement of (1.3) immediately.

Corollary 3.3. If , then where where

Corollary 3.4. If , then where Since , it is known from Lemma 2.3 that .
If and in (3.24) are replaced by zero, then the inequality (3.24) can be reduced to The inequalities (3.24) and (3.26) are refinements of the Hilbert-Ingham inequality

One has yet a new inequality according to Theorem 3.1(ii).

Theorem 3.5. With the assumptions as Theorem 3.1, if , then where

4. Applications to Function

Let be analytic in the unit-disc . If with then where the coefficient is the best possible. It is called the Fejer-Riesz inequality in function [7].

We will give both an extension and a refinement of (4.1) in what follows.

Theorem 4.1. Let where and . If is analytic in the unit-disc , then where .

Proof. At first, we prove the theorem for case Let . It is easy to deduce that By Lemma 2.4, we have Since the series is absolutely convergent, it is justified that the complex number is replaced by
According (3.18), we have where is defined by (3.19). It is easy to deduce that Because , . It shows that . Hence, the inequality (4.2) is valid when . If , then by the Blaschke decomposition theorem, it holds that , where is Blaschke function and in and
Let . According to the above result for we have Based on the case for , we have . Hence, The proof of Theorem 4.1 is completed.
Let Then, It is called the Hardy inequality in function [7].

We will give both an extension and a refinement of (4.9) as follows.

Theorem 4.2. Let be analytic in the unit-disc , where (with and ) and . Then, where

Proof. By Blaschke decomposition theorem, we have where is Blaschke function, . Let It is easy deduce that Owing to , it holds that . It is easy to deduce that By Lemma 2.4, we find that It follows from (4.13), (4.14), (4.15), and Corollary 3.2 that where is defined by (3.19). It is easy to deduce that These show that the inequality (4.10) is valid.

Acknowledgment

The research is supported by the Scientific Research Fund of Hunan Provincial Education Department (no. 06C657).

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