Advanced Inequalities

by George A Anastassiou

Overview

This monograph presents univariate and multivariate classical analyses of advanced inequalities. This treatise is a culmination of the author's last thirteen years of research work. The chapters are self-contained and several advanced courses can be taught out of this book. Extensive background and motivations are given in each chapter with a comprehensive list of references given at the end.The topics covered are wide-ranging and diverse. Recent advances on Ostrowski type inequalities, Opial type inequalities, Poincare and Sobolev type inequalities, and Hardy-Opial type inequalities are examined. Works on ordinary and distributional Taylor formulae with estimates for their remainders and applications as well as Chebyshev-Gruss, Gruss and Comparison of Means inequalities are studied.The results presented are mostly optimal, that is the inequalities are sharp and attained. Applications in many areas of pure and applied mathematics, such as mathematical analysis, probability, ordinary and partial differential equations, numerical analysis, information theory, etc., are explored in detail, as such this monograph is suitable for researchers and graduate students. It will be a useful teaching material at seminars as well as an invaluable reference source in all science libraries.

Product Details

ISBN-13:	9789814317627
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	10/27/2010
Series:	Series On Concrete And Applicable Mathematics , #11
Pages:	424
Product dimensions:	6.30(w) x 9.80(h) x 1.10(d)

Table of Contents

Preface vii

1 Introduction 1

2 Advanced Univariate Ostrowski Type Inequalities 5

2.1 Introduction 5

2.2 Auxilliary Results 6

2.3 Main Results 14

3 Higher Order Ostrowski Inequalities 21

3.1 Introduction 21

3.2 Main Results 22

4 Multidimensional Euler Identity and Optimal Multidimensional Ostrowski Inequalities 27

4.1 Introduction 27

4.2 Background 30

4.3 Main Results 31

4.4 Applications 53

4.5 Sharpness 76

5 More on Multidimensional Ostrowski Type Inequalities 81

5.1 Introduction 81

5.2 Auxilliary Results 82

5.3 Main Results 87

6 Ostrowski Inequalities on Euclidean Domains 93

6.1 Introduction 93

6.2 Main Results 93

7 High Order Ostrowski Inequalities on Euclidean Domains 99

7.1 Introduction 99

7.2 Main Results 99

7.3 Functions on General Domains 106

8 Ostrowski Inequalities on Spherical Shells 109

8.1 Introduction 109

8.2 Main Results 109

8.3 Addendum 123

9 Ostrowski Inequalities on Balls and Shells Via Taylor-Widder Formula 125

9.1 Introduction 125

9.2 Background 126

9.3 Results on the Shell 128

9.4 Results on the Sphere 132

9.5 Addendum 138

10 Multivariate Opial Type Inequalities for Functions Vanishing at an Interior Point 139

10.1 Introduction 139

10.2 Main Results 140

11 General Multivariate Weighted Opial Inequalities 149

11.1 Introduction 149

11.2 Main Results 151

12 Opial Inequalities for Widder Derivatives 161

12.1 Introduction 161

12.2 Background 161

12.3 Results 162

13 Opial Inequalities for Linear Differential Operators 171

13.1 Background 171

13.2 Results 171

14 Opial Inequalities for Vector Valued Functions 179

14.1 Introduction 179

14.2 Background 179

14.3 Results 180

14.4 Applications 186

15 Opial Inequalities for Semigroups 187

15.1 Introduction 187

15.2 Background 187

15.3 Results 189

16 Opial Inequalities for Cosine and Sine Operator Functions 197

16.1 Introduction 197

16.2 Background 197

16.3 Results 199

16.4 Applications 206

17 Poincaré Like Inequalities for Linear Differential Operators 209

17.1 Background 209

17.2 Results 210

18 Poincaré and Sobolev Like Inequalities for Widder Derivatives 215

18.1 Background 215

18.2 Results 217

19 Poincaré and Sobolev Like Inequalities for Vector Valued Functions 229

19.1 Introduction 229

19.2 Background 230

19.3 Results 231

19.4 Applications 240

20 Poincaré Type Inequalities for Semigroups, Cosine and Sine Operator Functions 243

20.1 Introduction 243

20.2 Semigroups Background 243

20.3 Poincaré Type Inequalities for Semigroups 245

20.4 Cosine and Sine Operator Functions Background 249

20.5 Poincaré Type Inequalities for Cosine and Sine Operator Functions 250

21 Hardy-Opial Type Inequalities 261

21.1 Results 261

22 A Basic Sharp Integral Inequality 271

22.1 Introduction 271

22.2 Results 271

23 Estimates of the Remainder in Taylor's Formula 275

23.1 Introduction 275

23.2 Some New Bounds for the Remainder 276

23.3 Some Further Bounds of the Remainder 280

23.4 Some Inequalities for Special Cases 282

23.5 Taylor-Multivariate Case Estimates 284

24 The Distributional Taylor Formula 293

24.1 Introduction and Background 293

24.2 Main Results 296

24.3 Applications 302

25 Chebyshev-Grüss Type Inequalities Using Euler Type and Fink Identities 305

25.1 Background 305

25.2 Main Results 307

26 Grüss Type Multivariate Integral Inequalities 319

26.1 Introduction 319

26.2 Auxiliary Result 320

26.3 Main Results 321

27 Chebyshev-Grüss Type Inequalities on Spherical Shells and Balls 331

27.1 Introduction 331

27.2 Main Results 332

28 Multivariate Chebyshev-Grüss and Comparison of Integral Means Inequalities 341

28.1 Background 341

28.2 Main Results 344

29 Multivariate Fink Type Identity Applied to Multivariate Inequalities 365

29.1 Introduction 365

29.2 Main Results 366

29.3 Applications 381

Bibliography 395

List of Symbols 407

Index 409