Ostrowski Type Inequalities and Applications in Numerical Integration

integral Inequalities Involving Functions With Bounded Derivatives, Otherwise Known As Ostrowski-type Integral Inequalities, Have Enjoyed A Surge In Popularity. This Field Has Developed Significantly Over The Last Few Years, And Has Yielded Many New Results And Powerful Applications In Numerical Integration, Probability Theory And Stochastics, Statistics, Information Theory, And Integral Operator Theory. the Main Aim Of The Present Work Is To Present A Number Of Selected Results On Ostrowski-type Integral Inequalities. Results For Univariate And Multivariate Real Functions And Their Natural Applications In The Error Analysis Of Numerical Quadratures For Both Simple And Multiple Integrals As Well As For The Riemann-stieltjes Integral Are Given. Topics Dealt With Include Generalisations Of The Ostrowski Inequality And Its Applications; Integral Inequalities For N-times Differentiable Mappings; Three-point Quadrature Rules; Product-branched Peano Kernels And Numerical Integration; Ostrowski-type Inequalities For Multiple Integrals; Results For Double Integrals Based On An Ostrowski-type Inequality; Product Inequalities And Weighted Quadrature; And Some Inequalities For The Riemann-stieltjes Integral. this Book Is Intended For Researchers And Graduate Students Working In The Fields Of Integral Inequalities, Approximation Theory, Applied Mathematics, Probability Theory And Stochastics, And Numerical Analysis. Preface......Page 5 1.1. Introduction......Page 9 1.2. Generalisations for Functions of Bounded Variation......Page 11 1.3. Generalisations for Functions whose Derivatives are in L......Page 23 1.4. Generalisation for Functions whose Derivatives are in Lp......Page 36 1.5. Generalisations in Terms of L1-norm......Page 50 Bibliography......Page 59 2.1. Introduction......Page 63 2.2. Integral Identities......Page 64 2.3. Integral Inequalities......Page 72 2.4. The Convergence of a General Quadrature Formula......Page 79 2.5. Grüss Type Inequalities......Page 83 2.6. Some Particular Integral Inequalities......Page 88 2.7. Applications for Numerical Integration......Page 112 Bibliography......Page 126 3.1. Introduction......Page 129 3.2. Bounds Involving at most a First Derivative......Page 131 3.3. Bounds for n-Time Differentiable Functions......Page 194 Bibliography......Page 218 4.1. Introduction......Page 223 4.2. Fundamental Results......Page 225 4.3. Simpson Type Formulae......Page 232 4.4. Perturbed Results......Page 234 4.5. More Perturbed Results Using -Seminorms......Page 243 Bibliography......Page 249 5.1. Introduction......Page 253 5.2. An Ostrowski Type Inequality for Double Integrals......Page 257 5.3. Other Ostrowski Type Inequalities......Page 271 5.4. Ostrowski's Inequality for Hölder Type Functions......Page 281 Bibliography......Page 288 6.1. Introduction......Page 291 6.3. Mapping Whose First Derivatives Belong to L(a,b)......Page 292 6.4. Numerical Results......Page 297 6.5. Application For Cubature Formulae......Page 298 6.6. Mapping Whose First Derivatives Belong to Lp(a,b).......Page 301 6.7. Application For Cubature Formulae......Page 304 6.8. Mappings Whose First Derivatives Belong to L1(a,b).......Page 306 6.9. Integral Identities......Page 309 6.10. Some Integral Inequalities......Page 313 6.11. Applications to Numerical Integration......Page 320 Bibliography......Page 322 7.1. Introduction......Page 325 7.2. Weight Functions......Page 326 7.3. Weighted Interior Point Integral Inequalities......Page 327 7.4. Weighted Boundary Point (Lobatto) Integral Inequalities......Page 340 7.5. Weighted Three Point Integral Inequalities......Page 347 Bibliography......Page 358 8.1. Introduction......Page 361 8.2. Some Trapezoid Like Inequalities for Riemann-Stieltjes Integral......Page 363 8.3. Inequalities of Ostrowski Type for the Riemann-Stieltjes Integral......Page 380 8.4. Some Inequalities of Grüss Type for Riemann-Stieltjes Integral......Page 400 Bibliography......Page 406 It was noted in the preface of the book'Inequalities Involving Functions and Their Integrals and Derivatives', Kluwer Academic Publishers, 1991, by D.S. Mitrinovic, J.E. Pecaric and A.M. Fink; since the writing of the classical book by Hardy, Littlewood and Polya (1934), the subject of differential and integral inequalities has grown by about 800%. Ten years on, we can confidently assert that this growth will increase even more significantly. Twenty pages of Chapter XV in the above mentioned book are devoted to integral inequalities involving functions with bounded derivatives, or, Ostrowski type inequalities. This is now itself a special domain of the Theory of Inequalities with many powerful results and a large number of applications in Numerical Integration, Probability Theory and Statistics, Information Theory and Integral Operator Theory. The main aim of the present book, jointly written by the members of the Vic­ toria University node of RGMIA (Research Group in Mathematical Inequali­ ties and Applications, http: I /rgmia. vu. edu. au) and Th. M. Rassias, is to present a selected number of results on Ostrowski type inequalities. Results for univariate and multivariate real functions and their natural applications in the error analysis of numerical quadrature for both simple and multiple integrals as well as for the Riemann-Stieltjes integral are given.