On a New Weighted Hilbert Inequality
© He Leping et al. 2008
Received: 13 January 2008
Accepted: 18 May 2008
Published: 20 May 2008
It is shown that a weighted Hilbert inequality for double series can be established by introducing a proper weight function. Thus, a quite sharp result of the classical Hilbert inequality for double series is obtained. And a similar result for the Hilbert integral inequality is also proved. Some applications are considered.
where , and the constant factor in (1.2) is the best possible value. Recently, various improvements and extensions of (1.1) and (1.2) appear in a great deal of papers (see [2–6], etc.). The aim of the present paper is to build some new inequalities by using the weight function method and the technique of analysis, and then to study some applications of them.
First we give some lemmas.
where is an arbitrary constant. This result is given in the paper (see ). Based on this indefinite integral it is easy to deduce that the equality (1.3) holds.
At first, notice that , hence we can write in the form of: , where and . It is obvious that the functions and are monotonously decreasing in . So, is also monotonously decreasing in . In the next place, notice that , therefore we can write in the form of: , where and . It is clear that the functions and are monotonously decreasing in . So, is also monotonously decreasing in .
By Lemma 1.1, we obtain the first integral of (1.5) at once after some simple computations and simplifications.
Similarly, the second integral of (1.5) can be gotten.
Its proof is similar to that of Lemma 1.3. Hence, it is omitted.
2. Main Results
And then by using the inequality (2.8), the inequality (2.1) follows from (2.9) at once. It is obvious that the inequality (2.1) is a refinement of (1.1). Below, we give an extension of (1.2).
It follows from (2.19) and (2.20) that the inequality (2.10) is valid.
As applications, we will give some new refinements of Hardy-Littlewood's theorem and Widder's theorem below.
It is known from (2.8) and (3.4) that the inequality (3.3) is valid. Theorem is therefore proved.
This is the famous Widder theorem (see ).
This work is a project supported by Scientific Research Fund of Hunan Provincial Education Department (07C520 and 06C657).
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