In 1934, G. H. Hardy et al. published a book entitled “Inequalities”, in which a few theorems about Hilbert-type inequalities with homogeneous kernels of degree-one were considered. Since then, the theory of Hilbert-type discrete and integral inequalities is almost built by Prof. Bicheng Yang in their four published books.
This monograph deals with half-discrete Hilbert-type inequalities. By means of building the theory of discrete and integral Hilbert-type inequalities, and applying the technique of Real Analysis and Summation Theory, some kinds of half-discrete Hilbert-type inequalities with the general homogeneous kernels and non-homogeneous kernels are built. The relating best possible constant factors are all obtained and proved. The equivalent forms, operator expressions and some kinds of reverses with the best constant factors are given. We also consider some multi-dimensional extensions and two kinds of multiple inequalities with parameters and variables, which are some extensions of the two-dimensional cases. As applications, a large number of examples with particular kernels are also discussed.
The authors have been successful in applying Hilbert-type discrete and integral inequalities to the topic of half-discrete inequalities. The lemmas and theorems in this book provide an extensive account of these kinds of inequalities and operators. This book can help many readers make good progress in research on Hilbert-type inequalities and their applications.

Preface v

Acknowledgments ix

1. Recent Developments of Hilbert-Type Inequalities with Applications 1

1.1 Introduction 1

1.2 Hilbert’s Inequality and Hilbert’s Operator 2

      1.2.1 Hilbert’s Discrete and Integral Inequalities 2

      1.2.2 Operator Formulation of Hilbert’s Inequality 4

      1.2.3 A More Accurate Discrete Hilbert’s Inequality 5

      1.2.4 Hilbert’s Inequality with One Pair of Conjugate Exponents 6

      1.2.5 A Hilbert-type Inequality with the General Homogeneous Kernel of Degree −1 9

      1.2.6 Two Multiple Hilbert-type Inequalities with the Homogeneous Kernels of Degree (−n + 1) 12

1.3 Modern Research for Hilbert-type Inequalities 12

      1.3.1 Modern Research for Hilbert’s Integral Inequality 12

      1.3.2 On the Way of Weight Coefficient for Giving a Strengthened Version of Hilbert’s Inequality 14

      1.3.3 Hilbert’s Inequality with Independent Parameters 15

      1.3.4 Hilbert-type Inequalities with Multi-parameters 18

1.4 Some New Applications for Hilbert-type Inequalities 22

      1.4.1 Operator Expressions of Hilbert-type Inequalities 22

      1.4.2 Some Basic Hilbert-type Inequalities 23

      1.4.3 Some Applications to Half-discrete Hilbert-type Inequalities 25

1.5 Concluding Remarks 27

2. Improvements of the Euler-Maclaurin Summation Formula and Applications 29

2.1 Introduction 29

2.2 Some Special Functions Relating Euler-Maclaurin’s Summation Formula 29

      2.2.1 Bernoulli’s Numbers 29

      2.2.2 Bernoulli’s Polynomials 31

      2.2.3 Bernoulli’s Functions 32

      2.2.4 The Euler-Maclaurin Summation Formula 34

2.3 Estimations of the Residue Term about a Class Series 36

      2.3.1 An Estimation under the More Fortified Conditions 36

      2.3.2 Some Estimations under the More Imperfect Conditions 40

      2.3.3 Estimations of δq(m, n) and Some Applications 47

2.4 Two Classes of Series Estimations 50

      2.4.1 One Class of Convergent Series Estimation 50

      2.4.2 One Class of Finite Sum Estimation on Divergence Series 52

3. A Half-Discrete Hilbert-Type Inequality with a General Homogeneous Kernel 57

3.1 Introduction 57

3.2 Some Preliminary Lemmas 58

      3.2.1 Definition of Weight Functions and Related Lemmas 58

      3.2.2 Estimations about Two Series 62

      3.2.3 Some Inequalities Relating the Constant k(λ1) 68

3.3 Some Theorems and Corollaries 70

      3.3.1 Equivalent Inequalities and their Operator Expressions 70

      3.3.2 Two Classes of Equivalent Reverse Inequalities 76

      3.3.3 Some Corollaries 82

      3.3.4 Some Particular Examples 97

      3.3.5 Applying Condition (iii) and Corollary 3.8 101

      3.3.6 Applying Condition (iii) and Corollary 3.4 115

4. A Half-Discrete Hilbert-Type Inequality with a Non-Homogeneous Kernel 123

4.1 Introduction 123

4.2 Some Preliminary Lemmas 123

      4.2.1 Definition of Weight Functions and Some Related Lemmas 123

      4.2.2 Estimations of Two Series and Examples 128

      4.2.3 Some Inequalities Relating the Constant k(α) 132

4.3 Some Theorems and Corollaries 134

      4.3.1 Equivalent Inequalities and their Operator Expressions 134

      4.3.2 Two Classes of Equivalent Reverses 140

      4.3.3 Some Corollaries 146

4.4 Some Particular Examples 155

      4.4.1 Applying Condition (i) and Corollary 4.5 155

      4.4.2 Applying Condition (iii) and Corollary 4.2 160

5. Multi-dimensional Half-Discrete Hilbert-Type Inequalities 169

5.1 Introduction 169

5.2 Some Preliminary Results and Lemmas 170

      5.2.1 Some Related Lemmas 170

      5.2.2 Some Results about the Weight Functions 172

      5.2.3 Two Preliminary Inequalities 175

5.3 Some Inequalities Related to a General Homogeneous Kernel 178

      5.3.1 Several Lemmas 178

      5.3.2 Main Results 183

      5.3.3 Some Corollaries 192

      5.3.4 Operator Expressions and Some Particular Examples 196

5.4 Some Inequalities Relating a General Non-Homogeneous Kernel 205

      5.4.1 Some Lemmas 205

      5.4.2 Main Results 210

      5.4.3 Some Corollaries 219

      5.4.4 Operator Expressions and Some Particular Examples 223

6. Multiple Half-Discrete Hilbert-Type Inequalities 233

6.1 Introduction 233

6.2 First Kind of Multiple Hilbert-type Inequalities 234

      6.2.1 Lemmas Related to the Weight Functions 234

      6.2.2 Two Preliminary Inequalities 245

      6.2.3 Main Results and Operator Expressions 248

      6.2.4 Some Kinds of Reverse Inequalities 253

6.3 Second Kind of Multiple Hilbert-type Inequalities 259

      6.3.1 Lemmas Related to the Weight Functions 259

      6.3.2 Two Preliminary Inequalities 269

      6.3.3 Main Results and Operator Expressions 271

      6.3.4 Some Kinds of Reverse Inequalities 276

6.4 Some Examples with the Particular Kernels 281

      6.4.1 The Case of kλ(x1, ··· , xm, xm+1) = 1(m+1i=1 xi)λ 282

      6.4.2 The Case of kλ(x1, ··· , xm+1) = sk=11mi=1 xiλ/s+ckxλ/sm+1 291

      6.4.3 The Case of kλ(x1, ··· , xm, xm+1) = 1(max1≤i≤m+1{xi})λ 302

      6.4.4 The Case of kλ(x1, ··· , xm, xm+1) = 1(min1≤i≤m+1{xi})λ 311

Bibliography 321

Index 331

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