Ostrowski Type Inequalities and Applications in Numerical Integration

Contents

Preface  V
Chapter 1. Generalisations of Ostrowski Inequality and Applications  1
1.1. Introduction 1
1.2. Generalisations for Functions of Bounded Variation  3
1.3. Generalisations for Functions whose Derivatives are in L 15
1.4. Generalisation for Functions whose Derivatives are in Lp  28
1.5. Generalisations in Terms of L1-norm  42
Bibliography 53
Chapter 2. Integral Inequalities for n -Times Differentiable Mappings 55
2.1. Introduction 55
2.2. Integral Identities 56
2.3. Integral Inequalities 64
2.4. The Convergence of a General Quadrature Formula  71
2.5. Grüss Type Inequalities  75
2.6. Some Particular Integral Inequalities  80
2.7. Applications for Numerical Integration 104
2.8. Concluding Remarks 118
Bibliography 119
Chapter 3. Three Point Quadrature Rules  121
3.1. Introduction  121
3.2. Bounds Involving at most a First Derivative  123
3.3. Bounds for n -Time Differentiable Functions  186
Bibliography  211
Chapter 4. Product Branches of Peano Kernels and Numerical Integration  215
4.1. Introduction  215
4.2. Fundamental Results  217
4.3. Simpson Type Formulae  224
4.4. Perturbed Results  226
4.5. More Perturbed Results Using d -Seminorms 235
4.6. Concluding Remarks 241
Bibliography  243
Chapter 5. Ostrowski Type Inequalities for Multiple Integrals 245
5.1. Introduction 245
5.2. An Ostrowski Type Inequality for Double Integrals  249
5.3. Other Ostrowski Type Inequalities 263
5.4. Ostrowski  Inequality for Hölder Type Functions 273
Bibliography 281
Chapter 6. Some Results for Double Integrals Based on an Ostrowski Type Inequality  283
6.1. Introduction 283
6.2. The One Dimensional Ostrowski Inequality 284
6.3. Mapping Whose First Derivatives Belong to L(a,b ) 284
6.4. Numerical Results  289
6.5. Application For Cubature Formulae  290
6.6. Mapping Whose First Derivatives Belong to Lp (a,b ). 293
6.7. Application For Cubature Formulae  296
6.8. Mappings Whose First Derivatives Belong to L1(a,b ). 298
6.9. Integral Identities  301
6.10. Some Integral Inequalities  305
6.11. Applications to Numerical Integration  312
Bibliography  315
Chapter 7.  Product Inequalities and Weighted Quadrature  317
7.1. Introduction  317
7.2. Weight Functions  318
7.3. Weighted Interior Point Integral Inequalities 319
7.4. Weighted Boundary Point (Lobatto)Integral Inequalities 332
7.5. Weighted Three Point Integral Inequalities 339
Bibliography 351
Chapter 8. Some Inequalities for Riemann-Stieltjes Integral 353
8.1. Introduction 353
8.2. Some Trapezoid Like Inequalities for Riemann-Stieltjes Integral 355
8.3. Inequalities of Ostrowski Type for the Riemann-Stieltjes Integral 372
8.4. Some Inequalities of Grüss Type for Riemann-Stieltjes Integral  392
Bibliography 401