# On a Multiple Hilbert's Inequality with Parameters

- Qiliang Huang
^{1}Email author

**2010**:309319

https://doi.org/10.1155/2010/309319

© Qiliang Huang. 2010

**Received: **12 May 2010

**Accepted: **31 August 2010

**Published: **2 September 2010

## Abstract

By introducing multiparameters and conjugate exponents and using Hadamard's inequality and the way of real analysis, we estimate the weight coefficients and give a multiple more accurate Hilbert's inequality, which is an extension of some published results. We also prove that the constant factor in the new inequality is the best possible and consider its equivalent form.

## 1. Introduction

where the constant factor is the best possible. Inequalities (1.1)–(1.4) are important in analysis and their applications (cf. [3]). In near one century, there are many improvements, generalizations and, applications of (1.1)–(1.4) in numerous literatures and monographs of mathematics (cf. [2–18]). Yang and Huang also considered the multiple Hilbert-type integral inequality (cf. [19, 20]). Recently, Yang summarized the methods of introducing parameters and estimating the weight coefficients to extend Hilbert-type inequalities for the past 100 years. Some representative results are as follows (cf. [21, 22]):

The constant factors in the above five inequalities are all the best possible. Inequalities (1.5) and (1.7) are generalizations of inequality (1.2), and inequality (1.9) is a multiple extension of (1.1). Inequalities (1.6) and (1.8) are the equivalent forms of (1.5) and (1.7), which are extensions of (1.4).

In this paper, by introducing multi-parameters and conjugate exponents and using Hadamard's inequality, we estimate the weight coefficients and give a multiple more accurate Hilbert 's inequality, which is an extension of inequalities (1.5), (1.7), and (1.9). We also prove that the constant factor in the new inequality is the best possible and consider its equivalent form.

## 2. Some Lemmas

Lemma 2.1.

Proof.

and then (2.1) is valid.

Lemma 2.2.

Proof.

Hence, we prove that the left-hand side of (2.3) is valid.

Lemma 2.3.

Proof.

and then (2.11) is valid by using inequality (2.3).

Hence, (2.12) is valid.

## 3. Main Results

Theorem 3.1.

Proof.

Note that for by (3.5), we directly get (3.6). Hence, (3.3) is valid by (3.6) and (2.12).

Then (3.3) is valid, which is equivalent to (3.2).

Theorem 3.2.

Let the assumptions of Theorem 3.1 be fulfilled, then the same constant factor in (3.2) and (3.3) is the best possible.

Proof.

For , we have Hence, is the best value of (3.2).

We conform that the constant factor in (3.3) is the best possible, otherwise we can get a contradiction by (3.7) that the constant factor in (3.2) is not the best possible.

## Declarations

### Acknowledgments

This work is supported by the Emphases Natural Science Foundation of Guangdong Institution, Higher Learning, College and University (no. 05Z026), and Guangdong Natural Science Foundation (no. 7004344).

## Authors’ Affiliations

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