1. Introduction

In 1934, Hardy published the following theorem (cf. [1, Theorem 319]).

Theorem A.

If is a homogeneous function of degree in , , , and , then for , , and , one has

(1.1)

where the constant factor is the best possible.

Hardy [2] also published the following Hardy's integral inequality.

Theorem B.

If , , , and ; , , then one has

(1.2)

where the constant factor is the best possible (cf. [1, Theorem  330]).

In 2009, Yang [3] published the following theorem.

Theorem C.

If , , , is a homogeneous function of degree in , and for any , , then for , , , , , and , we have

(1.3)

where the constant factor is the best possible.

For, , (1.3) reduces to (1.1). We name of (1.1) and (1.3) Hilbert-type integral inequalities. Inequalities (1.1), (1.2) and (1.3) are important in analysis and its applications (cf. [4–6]).

Setting , , , , by applying (1.2) (for ), Das and Sahoo gave a new integral inequality similar to Pachpatte's inequality (cf. [7, 8]) as follows:

(1.4)

where the constant factor is the best possible (cf. [9]). Sulaiman [10] also considered a Hilbert-Hardy-type integral inequality similar to (1.4) with the kernel , , . But he cannot show that the constant factor in the new inequality is the best possible.

In this paper, by applying the way of weight functions and inequality (1.2) for , a Hilbert-Hardy-type integral operator is defined, and the norm of operator is obtained. As applications, a new Hilbert-Hardy-type inequality similar to (1.3) is given, and two equivalent inequalities with a best constant factor as well as some particular examples are considered.

2. A Lemma and Two Equivalent Inequalities

Lemma 2.1.

If , is a nonnegative homogeneous function of degree in with , and for any , , then and

(2.1)

Proof.

Setting , we find

(2.2)

There exists , satisfying and . Since we find

(2.3)

there exists , such that , and then

(2.4)

The lemma is proved.

Theorem 2.2.

If , , , is a homogeneous function of degree in , and for any , , then for , , ,

(2.5)

, and , one has the following equivalent inequalities:

(2.6)
(2.7)

Proof.

Setting the weight functions and as follows:

(2.8)

then by Lemma 2.1, we find

(2.9)

By Hölder's inequality (cf. [11]) and (2.8), (2.9), we obtain

(2.10)

Then by Fubini theorem (cf. [12]), it follows:

(2.11)

Since , , then by (1.2) (for , we have

(2.12)

Hence by (2.11), we have (2.7). Still by Hölder's inequality, we find

(2.13)

Then by (2.7), we have (2.6).

On the other-hand, supposing that (2.6) is valid, by (2.11) and (1.2) (for , it follows . If , then (2.7) is naturally valid; if , setting

(2.14)

then by (2.6), we find

(2.15)

Hence, we have (2.7), which is equivalent to (2.6).

3. A Hilbert-Hardy-Type Integral Operator and Applications

Setting a real function space as follows:

(3.1)

for , , define an integral operator as follows:

(3.2)

Then, by (2.7), , and is bounded with

(3.3)

Theorem 3.1.

Let the assumptions of Theorem 2.2 be fulfilled, and additionally setting . Then one has

(3.4)

where the constant factor is the best possible. Moreover the constant factor in (2.6) and (2.7) is the best possible and then

(3.5)

Proof.

Since , by (1.2), for , it follows:

(3.6)

Then, by (2.6), we have (3.4).

For , setting as follows:

(3.7)

then for , , we find

(3.8)

where , , and are indicated as follows;

(3.9)

If there exists a positive constant , such that (3.4) is still valid as we replace by , then in particular, we find

(3.10)

By (3.8) and (3.10), we find

(3.11)

Since by Fubini theorem, we obtain

(3.12)

then for in (3.10), by Lemma 2.1, we obtain . Hence , and then is the best value of (3.4).

We conclude that the constant factor in (2.6) is the best possible, otherwise we can get a contradiction by (1.2) that the constant factor in (3.4) is not the best possible. By the same way, if the constant factor in (2.7) is not the best possible, then by (2.13), we can get a contradiction that the constant factor in (2.6) is not the best possible. Therefore in view of (3.3), we have (3.5).

Corollary 3.2.

For , , , , , in (2.6), (2.7) and (3.4), one has the following basic Hilbert-Hardy-type integral inequalities with the best constant factors:

(3.13)
(3.14)
(3.15)

where , and (3.13) is equivalent to (3.14).

Example 3.3.

For , , , , and in(3.4),

(a)if , , and , then we obtain the following integral inequalities:

(3.16)

(b)if, , then we have

(3.17)

(c)if , then we find

(3.18)

where the constant factors in the above inequalities are the best possible.