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International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 486127, 11 pages
http://dx.doi.org/10.1155/2010/486127
Research Article

A Hilbert Integral-Type Inequality with Parameters

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

Received 15 March 2010; Revised 9 April 2010; Accepted 14 April 2010

Academic Editor: Feng Qi

Copyright © 2010 Shang Xiaozhou 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

A Hilbert-type integral inequality with parameters and can be established by introducing a nonhomogeneous kernel function. And the constant factor is proved to be the best possible. And then some important and especial results are enumerated. As applications, some equivalent forms are studied.

1. Introduction

Let . Then where the constant factor is the best possible, and the equality in (1.1) holds if and only if , or .This is the famous Hilbert integral inequality (see [1, 2]). Owing to the importance of the Hilbert inequality in analysis and applications, some mathematicians have been studying them. Recently, various improvements and extensions of (1.1) appear in a great deal of papers (see [311], etc.). Specially, Gao and Hsu enumerated the research articles to more than 40 in the paper [6]. It is obvious that the integral kernel function of the left hand side of (1.1) is a homogeneous form of degree 1. In general, the Hilbert type integral inequality with a homogeneous kernel of degree 1 has been studied in the paper [1]. The purpose of the present paper is to establish a Hilbert integral type inequality with a non-homogeneous kernel. And the constant factor is proved to be the best possible. And then some important and especial results are enumerated and some equivalent forms are studied.

For convenience, we need to introduce the Catalan constant and define a real function.

The Catalan constant is defined by where is the complete elliptic integral, namely, And the approximation of is that These results can be found in the paper [12, page 503].

Let . Define a real function by

In particular, , where is the Catalan constant.

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

Lemma 1.1. Let be a positive number and . Then

Proof. According to the definition of the gamma function, we obtain immediately (1.6). This result can be also found in the paper [12, page 226, formula ].

Lemma 1.2. Let . Define a function by where is the gamma function, and is a function defined by (1.5). Then where .

Proof. It is easy to deduce that Expanding into power series of and then using Lemma 1.1, we have
It follows from (1.9), (1.10), and (1.7) that the equality (1.8) holds.

2. Main Results

In the section we will formulate our main results.

Theorem 2.1. Let and be two real functions, and let and be arbitrary two positive numbers. If and , then where is defined by (1.7). And the constant factor in (2.1) is the best possible. And the equality in (2.1) holds if and only if , or .

Proof. We may apply Hardy’s technique and Cauchy-Schwarz’s inequality to estimate the left-hand side of (2.1) as follows: where .
By substituting for , it is easy to deduce that Based on (1.8), we obtain It is known from (2.2) and (2.4) that the inequality (2.1) is valid.
If , or , then the equality in (2.1) obviously holds. If and , then
If (2.2) takes the form of the equality, then there exists a pair of non-zero constants and such that Then we have Without losing the generality, we suppose that ; then This contradicts that . Hence it is impossible to take the equality in (2.2). It shows that it is also impossible to take the equality in (2.1).
It remains to need only to show that in (2.1) is the best possible. Below we will apply Yang’s technique (see [13]) to verify our assertion.
For all , define two functions by Then we have
Let such that the inequality (2.1) is still valid when is replaced by . Specially, we have Let . By Fubini’s theorem, we have By Fatou’s lemma and (1.8), we have
It follows that in (2.1) is the best possible. Thus the proof of theorem is completed.

Based on Theorem 2.1, we may build some important and interesting inequalities.

Theorem 2.2. Let be a nonnegative integer and . If and , then where the are the Euler numbers, namely, and so forth, and the constant factor in (2.14) is the best possible. And the equality in (2.14) holds if and only if , or .

Proof. We need only to verify the constant factor in (2.14). When , it is known from (1.10) that
It is known from the paper [14] that where the are the Euler numbers, namely, and so forth.
It follows that the constant factor in (2.14) is correct.

In particular, based on Theorem 2.2, the following results are gotten.

Corollary 2.3. If and , then where the constant factor in (2.17) is the best possible. And the equality in (2.17) holds if and only if , or .

Corollary 2.4. If and , then where the constant factor in (2.18) is the best possible. And the equality in (2.18) holds if and only if , or .

Theorem 2.5. Let and be two real functions, and a positive number. If and , then where is the Catalan constant, that is, . And the constant factor in (2.19) is the best possible. And the equality in (2.19) holds if and only if , or .

Proof. We need only to verify the constant factor in (2.19).  For case , based on (1.7), (1.5) and (1.2), it is easy to deduce that the constant factor is that .

In particular, when , the following result is obtained

Corollary 2.6. Let and be two real functions. If and , then where is the Catalan constant, that is, . And the constant factor in (2.20) is the best possible. And the equality in (2.20) holds if and only if, or .

A great deal of the Hilbert type inequalities can be established provided that the parameters are properly selected

3. Some Equivalent Forms

As applications, we will build some new inequalities.

Theorem 3.1. Let be a real function, and let and be arbitrary two positive numbers. If , then where is defined by (1.7) and the constant factor in (3.1) is the best possible. And the equality in (3.1) holds if and only if And the inequality (3.1) is equivalent to (2.1).

Proof. First, we assume that the inequality (2.1) is valid. Define a function by By using (2.1), we have It follows from (3.3) that the inequality (3.1) is valid after some simplifications.
On the other hand, assume that the inequality (3.1) keeps valid; by applying in turn Cauchy’s inequality and (3.1), we have Therefore, the inequality (3.1) is equivalent to (2.1).
If the constant factor in (3.1) is not the best possible, then it is known from (3.4) that the constant factor in (2.1) is also not the best possible. This is a contradiction. The theorem is proved.

Theorem 3.2. Let be a nonnegative integer and . If , then where , the are the Euler numbers,namely, , and so forth, and the constant factor is the best possible. And the equality in (3.5) holds if and only if .  And the inequality (3.5) is equivalent to (2.14).

Theorem 3.3. Let and be two real functions, and let be a positive numbers. If and , then where is the Catalan constant, that is, . And the constant factor in (3.6) is the best possible. And the equality in (3.6) holds if and only if . And the inequality (3.6) is equivalent to (2.19).

The proofs of Theorems 3.2 and 3.3 are similar to one of Theorem 3.1; they are omitted here.

Similarly, we can establish also some new inequalities which they are, respectively, equivalent to the inequalities (2.17), (2.18) and (2.20). These are omitted here.

Acknowledgments

Authors would like to express their thanks to the referees for their valuable comments and suggestions. The research is supported by Scientific Research Fund of Hunan Provincial Education Department (09C789).

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