On a Multiple Hilbert's Inequality with Parameters

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

In 1908, Weyl published the following famous Hilbert's inequality (cf. [1]). If , and then

(1.1)

where the constant factor is the best possible. In 1934, Hardy proved the following more accurate Hilbert's inequality (cf. [2]):

(1.2)

where the constant factor is the best possible. For the equivalent forms of (1.1) and (1.2) are given as follows (cf. [2]):

(1.3)

(1.4)

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]):

(i)if , , then

(1.5)

(1.6)

(1.7)

(1.8)

• (ii) if ,,, then

(1.9)

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.

If , , , , , then

(2.1)

Proof.

We find the following:

(2.2)

and then (2.1) is valid.

Lemma 2.2.

If ,, ,,, then

(2.3)

Proof.

For fixedwe set

(2.4)

In virtue of and we find , Putting we have the following:

(2.5)

Since by the following Hadamard's inequality (cf. [5]):

(2.6)

it follows that

(2.7)

and then we have the right-hand side of (2.3). Since

(2.8)

and is strictly decreasing in , we get

(2.9)

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

Lemma 2.3.

As the assumption of Lemma 2.1, define the weight coefficients as

(2.10)

, then there exists such that

(2.11)

Moreover, for any it follows that

(2.12)

Proof.

We prove (2.11) by mathematical induction. For we set and satisfying Putting ,, we have the following:

(2.13)

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

Assuming that for (2.11) is valid, then for setting , by (2.3), we have the following:

(2.14)

Setting , , we find , By the assumption of induction, it follows that

(2.15)

(2.16)

where and

(2.17)

Setting by (2.16), we have the following:

(2.18)

and then by (2.15), (2.18), and mathematical induction, (2.11) is valid. Setting ,,,,, then we have the following:

(2.19)

Hence, (2.12) is valid.

3. Main Results

Theorem 3.1.

Suppose that ,, ,, , , , , such that

(3.1)

then one has the following equivalent inequalities:

(3.2)

(3.3)

Proof.

Since by (2.1) and Hölder's inequality (cf. [5]), we find that

(3.4)

(3.5)

For since by Hölder's inequality again in (3.5), we have the following:

(3.6)

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

Since by Hölder's inequality once again, it follows that

(3.7)

By (3.3), we have (3.2). On the other hand, assuming that (3.2) is valid, setting

(3.8)

then we find that

(3.9)

By (3.2), it follows that If then (3.3) is naturally valid. Suppose that by (3.2), we find that

(3.10)

Dividing out into two sides of (3.10), we have the following:

(3.11)

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.

By (2.11) and

(3.12)

there exists such that for ,

(3.13)

where Setting

(3.14)

we find that

(3.15)

If there exists a constant such that (3.2) is still valid as we replace by then in particular, we have the following:

(3.16)

In virtue of (3.15) and (3.16), it follows that

(3.17)

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.

Remarks 3.3 ..

• (i) When the assumption of two theorems becomes (ii) When , (3.2) reduces to (1.9). (iii) For , setting ,in (3.2), then we obtain (1.5). Setting , in (3.2), we get (1.7).