[Abstract] This paper first introduces the development history of inequality theory, then introduces discrete Hilbert inequality, introduces an elementary proof of Hilbert inequality, and finally makes a brief summary of the generalized form of Hilbert inequality.

[Keywords:] inequality theory Hilbert inequality elementary proof weight function

Abstract: In this article, we first introduce the history of inequality theory. Then the Hilbert inequality is introduced, and finally a series of forms of Hilbert inequality are summarized.

[Keywords:] elementary proof of weight function of Hilbert inequality in inequality theory

1 quotation

1. 1 topic background

As we all know, inequality theory plays an important role in mathematical theory and permeates all fields of mathematics, so it is necessary to have a clear understanding of the development history of inequality theory.

Since Hilbert inequality was put forward, many mathematicians have given various proofs. This paper introduces an elementary proof. At the same time, various generalized forms of Hilbert inequality are summarized.

1.2 Main contents of this article

The work of this paper mainly includes three aspects:

(1), introducing the development history of inequality theory.

(2) Introduce Hilbert inequality and give elementary proof.

(3) Summarize all kinds of promotion forms of Hilbert.

A brief history of inequality theory and Hilbert inequality

2. A brief history of1inequality theory

The study of mathematical inequality originated in European countries, and there is a huge research group in Eastern European countries, especially in the former Yugoslavia. At present, mathematical workers who are interested in inequality theory are all over the world.

There are two watershed events in the development history of mathematical inequality theory, namely1Chebyshev's paper published in 882 and1Hardy's speech when he was the chairman of the London Mathematical Society in 928. Hardy, Littlewood and Puglia's Preface to Inequality gave incisive views on the philosophy of inequality: generally speaking, elementary inequalities should have elementary proofs, which should be "internal" and should give proof of the establishment of equal signs. A.M. Fink believes that people should do their best to state and prove inequalities that cannot be generalized. Hardy thinks that basic inequalities are basic. Since the famous mathematician G.H. Hardy and the book Inequality written by J.E. Littlewood and G. Puglia was published by Cambridge University Press in 1934, the research on mathematical inequality theory and its application has officially appeared in BLACKPINK, becoming a new mathematical discipline. Since then, inequality is no longer a combination of scattered and isolated formulas, but has developed into a systematic scientific theory.

Since 1970s, an international academic conference on general inequality has been held in Germany every four years, and a special collection of conference papers has been published. Inequality theory is also one of the themes of the Third World Conference of Nonlinear Analysts (WCNA-2000) held in Italy in 2000. The 6th and 7th international conferences on nonlinear functional analysis and application held in Korea in 2000 and 20001,and ISAAC held in Dalian University of Technology, China in 2000, all took the theory of mathematical inequality as the main topic of the conference agenda. 200 1 International Conference on Inequality was held in t heWest University, Romania from July 9, 2000/KLOC-0 to June 4, 2000/KLOC-0.

Historically, China mathematicians have made important contributions in the field of inequality, including Hua, Fan Qitai, Lin Dongpo, Wang Zhonglie and Wang Zhonglie. In recent years, many mathematicians in China are active in the field of international mathematical inequality theory and its application, and have made unique contributions in related fields, which have attracted the attention and attention of their counterparts at home and abroad. Such as Professor Wang Wanlan, Professor Shi Huannan, Professor Yang, Professor Gao Mingzhe, Professor Zhang, Professor, etc.

Since 1980s, there has been an upsurge of studying inequality in China. In 1980s, a series of pioneering work by Professor Yang Lu and others in the study of geometric inequalities pushed the study of geometric inequalities in China to a climax. In terms of algebraic inequality, Professor Wang Wanlan's in-depth study of Van ky inequality has reached the international leading level. Professor Qi Feng and his research team have made a great deal of systematic and cutting-edge research results in average inequality and other inequalities. For the analysis of inequality, Professor Hooke published a paper "An Inequality and Some Applications" in China Science 198 1. Aiming at the defects of Holder's inequality, he put forward a brand-new inequality, which was called "an outstanding new inequality" by American mathematical critics, and now it is called Hooke's (HK) inequality. Professor Hooke made a systematic and in-depth study on this inequality and its application.

At present, the research on mathematical inequality theory and its application in China also has rich achievements. For example, Mr. Kuang Jichang's monograph "Common Inequality" has been published in the second edition, and it has been reprinted many times in just a few years because of the shortage of supply. This is a rare phenomenon for mathematical monographs. The second influential monograph is Inequalities in Matrix Theory co-authored by Wang Songgui and Jia Zhongzhong. In addition, there is an active inequality research group in China, which sponsored an internal exchange publication, Inequality Research Newsletter, with mathematician Yang Lu as consultant.

Hilbert inequality was put forward by Hilbert in his lecture on integral equation. Since then, many famous mathematicians such as Phil (192 1), Framcis, Littlewood (1928), Hardy (1920) and Hardy-Littlewood-Paulia (1926). 193 1), Owen (1930), Paulia and szegegbu, Shure (19 1 1), F. Wiener (19/). For this reason, Hardy et al. specially discussed Hilbert inequality and its similar situation and generalization in Chapter 9x of the document "1". Since 1990s, a large number of China scholars, such as Professor He Yang, have made remarkable achievements in the study of Hilbert inequality and its similar situation and extension. Because these results have important theoretical and practical significance, they have caused a series of extensive research, made various progress and published the results in many newspapers and magazines.

To sum up, mathematical inequality theory is full of vitality and prosperity.

2.2 Elementary Proof of Hilbert Inequality

Proposition 1 (Hilbert inequality) If the sum is a square summable real series, the double series converges, and

( 1)

The inequality is strictly established, and the equation is established if and only if the constant is zero, which is the best in the formula (1).

The proof of proposition one must apply two lemmas.

Lemma A has a pair for every positive number M.

& lt

Prove that the set points (0,0), (0,) and (,) are c, y, (n = 0, 1, 2,? ), s represents the area from point C to Y circle with the center at point C and radius, and is the intersection point of straight line C and the vertical line passing through this point (n = 1, 2,3,? )。 In addition, let represent the area of sector C (figure below 1).

The area of the representation, so, get.

=S= >

=

= ?

=

& gt

Therefore,

Now we can prove Hilbert inequality. commemorate

=

Applying Schwartz inequality, we get

Lemma 1 is applied to the above. Obviously, the final inequality is strictly true if and only if the sequence is constant to zero.

Prove that it cannot be replaced by a constant smaller than it.

Lemma 2 for each natural number m> 1, is

& gt- 。

It is proved that let represent straight line and straight line (n = 0, 1, 2,? M- 1), indicating the area of the sector (Figure 2 below),

Obviously there is.

= & lt

= +

= +

= +

Therefore, >-

It is proved that Hilbert inequality is an optimal constant, and the sequence: = =, when, = = 0, when >； is considered. Where k is a natural number, then

+ +

(By Lemma 2)

-( )

therefore

-

Therefore, it is the optimal constant in Hilbert inequality. At this point, the elementary proof of Hilbert inequality is completed.

2.3 the generalization of Hilbert inequality

Hilbert put forward inequality.

( 1)

(2)

Later, Hardy extended these results, and he got the following inequality.

(3)

(4)

Here, 0, += 1, p q >；; 1。 Inequalities (3) and (4) are called Hardy-Hilbert multiple series inequalities, and the equal sign holds if and only if the sum is constant to zero.

Over the years, many mathematicians have studied Hilbert inequality and obtained a series of results. Let's briefly review the course of these studies. This paper first introduces the results based on Hilbert's most primitive inequality, and then shows a series of results about Hardy-Hilbert inequality.

In 1990, L.C.Hsu and others carefully analyzed Hardy's original methods and techniques, introduced a weight function w(n)=, and obtained an improved inequality:

(5)

Soon, Hsu and Wang simplified the weight function to find the maximum possible value of θ that can make Equation (5) hold. Later, L.C Hsu and Gao Mingzhe used different methods to obtain the lower supremum of θ= 1.28 1+ and then the upper supremum λ(λ= 1.4603545+) of θ, thus solving the problem.

As for inequality (2), Gao Mingzhe improved it.

w(n)=？ (n)>0(n= 1，2，…).

Then the Euler formula is applied to estimate the weight function w:

w(n)≤，θ= 17/20

Similarly, some new results are obtained on Hardy-Hilbert inequality.

In the process of studying Hardy-Hilbert inequality (3), the value of the summation formula with parameter n is estimated, such as

In 1990, Hsu and Guo first introduced the weight function:

Inequality (3) is extended to

Then Xu and Gao Mingzhe improved the weight function. Two years later, Gao gave the exact form of the weight function:

Soon after, Yang got a lower bound, which means that he got a better result in terms of weight function:

C is Euler constant, and (1-c) is proved to be the best constant of inequality. An upper bound proved by Gao Mingzhe is:

ρ(t)=t-[t]- 1/2

And it is estimated that

If the inequality > no longer holds, the problem will be completely solved.

Regarding inequality (4), Yang got the following good results:

R = p, q and c are constants.

In 1998, Yang and Debnath gave another form of Hardy-Hilbert inequality with weighted function:

In addition to the above, Yang has the following achievements:

If you replace s(n, r) in the above expression with, you will get other results.

2/kloc-0 At the beginning of the century, Tan Li improved inequality (3) by introducing a formal weight coefficient.

If,

So,

In the middle? =ln2- 13/48+？ / 1920(0 & lt； ? & lt 1), which is the best constant independent of R ..

And get the following inference:

set up

,

When q is large enough, it exists.

centre

Introducing appropriate parameters will make the object of study and research more general, and it is also a common method. In this part, the generalized Hilbert inequality with parametric form is summarized.

Recently, Yang introduced parameters A, B and λ to extend the inequality (1), and he established the following new inequality:

& lt

1. B > 0，0 & lt； λ≤2, and B(p, q) is a β function with the best constant. Yang got the following results:

& lt

A, B, C>0, 0 < λ≤ 2, also proved to be the best.

Give a generalization of inequality (4), Yang and De bernat:

& lt,

Constant = is the best, where 2-min (p, q).

Recently, Kuang Jichang and De bernat gave the general form of Hardy-Hilbert inequality:

,

p & gt 1， 1/p+ 1/q= 1， 1/2 & lt； ? min(p，q)，

K(x, y) is a nonnegative degree of -t (t >; 0). If there is a fourth-order continuous WeChat service on (0, +∞), when n= 1, 2,3,4, m=0, 1, y? +

& lt+ =p，q

therefore

& lt,

In ...

= & gt0,

r=p，q .

In order to update, considering inequalities (3) and (4), Yang and Debnath established new inequalities with parameters A, B and λ:

A constant factor of 3 is the best. In particular,

( 1) λ= 1，A，B& gt； 0

(2) λ=2，A，B& gt； 0

(3) 2 minutes {p, q} & ltλ≤2, A=B= 1,

The above constant factors are all optimal.

By introducing the parameter λ in another way, Yang got the following results:

The constant factor π/(λsinπ/p) is the best. In particular,

( 1) λ= 1,

(2) p=q=λ=2，

The constant factors of the above inequalities are all optimal.

Thirdly, Kuang Qichang established a new general form of Hilbert inequality.

1/p+ 1/q= 1, for every positive integer n < +∞, N=+∞,

Definition:

If 1

If it is 0

Based on the above conclusions, some important inferences are obtained:

Inference 1 Assuming above, then

Inference 2 Assume that as mentioned above,

A similar definition, if 1

If it is 0

Inference 3,

Definition:

If it is 0

In particular, if the following inequality holds:

Re-generalization of the application of new inequality;

In 1992, Hooke established a formal inequality:

This is a new extension of Hilbert's inequality theory.

Hook used some basic inequalities he got to draw some good conclusions, such as

certificate

A is a real number.

In 1996, Hooke draws a general conclusion with the parameter λ. Especially, when λ= 1/2, there is

When λ= 1, there is

If λ≠0 and λ is a non-negative integer, Hu gives the following results:

This is a generalization of Hilbert inequality and Ingham inequality.

When λ is a positive integer, Hu gives

When λ ≠ 0, 1, 2, ..., Hu recently proved this point.

This is a generalization of Polya-Szego inequality.

1999, Gao Mingzhe obtained a new inequality by using positive definite matrix:

Using this inequality, a stronger new inequality is obtained:

Soon, he used this formula to prove the following inequality:

The function s(x) is defined as

At the beginning of the 20th century, Yao Jinbin used the improved Cauchy inequality to give Yang a result:

Made improvements.

For convenience, let's make the following symbolic assumptions:

w(n)=？ -? (noun)

Is a unit vector with the following properties:

,? ,? Linearly independent

He has the following results:

if

Then,

Define a function? for

= 1 When m=n= 1,

When m = 0? n，m？ n

2 1 At the beginning of the century, Yang Qiaoshun extended inequality (4) by using improved Holder inequality and weight function.

For convenience, some symbols are introduced:

if

therefore

centre

Define a function?

= 1, when m=n=0

=0, when, m and n are not both 0.

From this, we can also draw the following inferences:

if

therefore

centre

Particularly noteworthy is the popularity of Hooke,

Twenty years ago, Hooke established an important inequality:

Recently, he got a new inequality:

manufacture

If it is.

Then there is

Among them,

In particular, if, then

When p=2, the above is the generalization of Holder inequality. Obviously, using these conclusions to estimate inequality (1)-(4) will get some new results. We believe that more generalizations of Hilbert inequality will continue to appear in the future.

3 summary

This paper mainly introduces the development history of inequality theory and Hilbert inequality, and completes the following work:

Firstly, this paper reviews the history of the development of inequality theory and introduces the research and contribution of Chinese and foreign mathematicians in the development of inequality theory.

Secondly, the form of Hilbert inequality is introduced and the elementary proof is given.

Thirdly, the generalization of Hilbert inequality by mathematicians at home and abroad is summarized.

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