Abstract

It is shown that some new extensions of Hilbert's integral inequality with parameter can be established by introducing a proper weight function. In particular, when , a refinement of Hilbert's integral inequality is obtained. As applications, some new extensions of Widder's inequality and Hardy-Littlewood's inequality are given.

1. Introduction and Lemmas

Let . It is well known that the inequality of the form is called Hilbert’s integral inequality, where the coefficient is the best possible.

In [1], by introducing a parameter (), the following extension of (1.1) of the form was established.

Recently, various improvements and extensions of (1.1) appear in a great deal of papers (see [2]). The aim of this paper is to give some new improvements of (1.1) and (1.2) and then present some important applications.

We now introduce some notations that will be used throughout the paper.

Let and let be an integrable function in . Define such that for . We also define

Lemma 1.1. With the above-mentioned assumptions, one has where the weight function is defined by

Proof. By using the substitution , it is easy to deduce that where is a function defined by (1.5).
Similarly, we have From the above equations involving and (1.4) holds true.

Lemma 1.2. Let and . Then

Proof. The case was studied in [3], or can be obtained by using [4]. Next, consider the case . By the definition and properties of beta function, it is easy to deduce that

2. Theorem and Its Corollary

Theorem 2.1. Let and be two real functions such that and , where . Then where the weight function is defined by

Proof. First, assume . Let , .
Then the following holds: In fact, it is obvious that We need only to show that Let . Then Noting that and applying Schwarz’s inequality, we have It follows from (1.4) that where the weight function is defined by (1.5).
Let , where and . It is obvious that . By Lemma 1.2, it is easy to deduce that Substitute into (2.8) to obtain
Next, consider the case . By Schwarz’s inequality, we have Based on (2.10), it follows from (2.11) that the inequality (2.1) is valid at once. Theorem is proved.

The special case in Theorem 2.1 yields the following Hilbert’s integral inequality.

Corollary 2.2. If and , then where the weight function is defined by

Proof. It follows directly from the proof of Theorem 2.1 and so the details are omitted.

Remark 2.3. By setting in Corollary 2.2, (2.12) yields where the weight function is defined by (2.13).

Remark 2.4. For the case in Theorem 2.1, (2.1) becomes where the weight function is defined by

3. Some Applications

As applications, we will give some extensions and refinements of Widder’s inequality and Hardy-Littlewood’s inequality.

Let Then Inequality (3.1) is called Widder’s inequality (see [5]).

We will give an extension of (3.1) below.

Theorem 3.1. Under the above assumptions, if , then where is defined by (2.13).

Proof. First, observe that the following holds: Let . Then we have where . Using Remark 2.3 , the inequality (3.2) follows from (3.4) at once.
In particular, when , we obtain a refinement of (3.1).

Corollary 3.2. With the assumptions as Theorem 3.1, if , then where is defined by (2.13).

Let If , , then we have Hardy-Littlewood’s inequality (see [6]) of the form where is the best constant that keeps (3.6) valid. In [7], the inequality (3.6) was extended to the following inequality: where ,