A Hilbert's Inequality with a Best Constant Factor

Abstract

We give a new Hilbert's inequality with a best constant factor and some parameters.

1. Introduction

If , , such that and , then the well-known Hardy-Hilbert's inequality and its equivalent form are given by

(1.1)

(1.2)

where the constant factors are all the best possible [1]. It attracted some attention in the recent years. Actually, inequalities (1.1) and (1.2) have many generalizations and variants. Equation (1.1) has been strengthened by Yang and others ( including integral inequalities ) [2–11].

In 2006, Yang gave an extension of [2] as follows.

If , , such that , , then

(1.3)

is the Beta function.

In 2007 Xie gave a new Hilbert-type Inequality [3] as follows.

If , and the right of the following inequalities converges to some positive numbers, then

(1.4)

The main objective of this paper is to build a new Hilbert's inequality with a best constant factor and some parameters.

In the following, we always suppose that

(1), , ,

(2)both functions and are differentiable and strict increasing in and respectively,

(3) are strictly increasing in and respectively. is strict decreasing on and ,

(4)

2. Some Lemmas

Lemma 2.1.

Define the weight coefficients as follows:

(2.1)

(2.2)

(2.3)

(2.4)

then

(2.5)

where

(2.6)

Proof.

Let then if then

(2.7)

if ), then

(2.8)

On the other hand, . Setting , then Similarly, .

Lemma 2.2.

For one has

(2.9)

Proof.

(2.10)

The lemma is proved.

Lemma 2.3.

Setting (or and (or , resp.), then is strictly decreasing, then

(2.11)

There for any ).

Proof.

We have

(2.12)

Easily, had up bounded when

3. Main Results

Theorem 3.1.

If , , , then

(3.1)

(3.2)

is defined by Lemma 2.1.

Proof.

By Hölder's inequality [12] and (2.5),

(3.3)

setting By(3.1) we have

(3.4)

By and (3.4) taking the form of strict inequality, we have (3.1). By Hölder's inequality[12], we have

(3.5)

as . By (3.2), (3.5) taking the form of strict inequality, we have (3.1).

Theorem 3.2.

If , then both constant factors, and of (3.1) and (3.2), are the best possible.

Proof.

We only prove that is the best possible. If the constant factor in (3.1) is not the best possible, then there exists a positive (with ), such that

(3.6)

For , setting , then

(3.7)

On the other hand and ),

(3.8)

By (3.6), (3.7), (3.8), and Lemma 2.3, we have

(3.9)

(3.10)

We have , . This contracts the fact that .