Frontiers In Time Scales And Inequalities

by George A Anastassiou

Overview

This monograph contains the author's work of the last four years in discrete and fractional analysis. It introduces the right delta and right nabla fractional calculus on time scales and continues with the right delta and right nabla discrete fractional calculus in the Caputo sense. Then, it shows representation formulae of functions on time scales and presents Ostrowski type inequalities, Landau type inequalities, Grüss type and comparison of means inequalities, all these over time scales. The volume continues with integral operator inequalities and their multivariate vectorial versions using convexity of functions, again all these over time scales. It follows the Grüss and Ostrowski type inequalities involving s-convexity of functions; and also examines the general case when several functions are involved. Then, it presents the general fractional Hermite-Hadamard type inequalities using m-convexity and (s, m)-convexity. Finally, it introduces the reduction method in fractional calculus and its connection to fractional Ostrowski type inequalities is studied.This book's results are expected to find applications in many areas of pure and applied mathematics, especially in difference equations and fractional differential equations. The chapters are self-contained and can be read independently, and advanced courses can be taught out of it. It is suitable for researchers, graduate students, seminars of the above subjects, and serves well as an invaluable resource for all science libraries.

Product Details

ISBN-13:	9789814704434
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	10/06/2015
Series:	Series On Concrete And Applicable Mathematics , #17
Pages:	288
Product dimensions:	6.60(w) x 9.80(h) x 0.80(d)

Table of Contents

Preface v

1 Foundations of Right Delta Fractional Calculus on Time Scales 1

1.1 Background 1

1.2 Results 3

Bibliography 9

2 Principles of Right Nabla Fractional Calculus on Time Scales 11

2.1 Background 11

2.2 Results 13

Bibliography 19

3 About Right Delta Discrete Fractionality 21

3.1 Introduction and Background 21

3.2 Main Results 25

Bibliography 31

4 About Right Nabla Discrete Fractional Calculus 33

4.1 Introduction and Background 33

4.2 Main Results 38

Bibliography 45

5 Representations and Ostrowski Inequalities over Time Scales 47

5.1 Introduction 47

5.2 Main Results 48

5.3 Applications 81

Bibliography 87

6 Landau Inequalities on Time Scales 89

6.1 Introduction 89

6.2 Background 89

6.3 Main Results 91

6.4 Applications 93

6.5 Addendum 94

Bibliography 97

7 Grüss and Comparison of Means Inequalities over Time Scales 99

7.1 Introduction 99

7.2 Main Results 100

7.3 Applications 119

Bibliography 123

8 About Integral Operator Inequalities over Time Scales 125

8.1 Introduction 125

8.2 Main Results 130

Bibliography 151

9 About Vectorial Integral Operator Inequalities Using Couvexity over Time Scales 153

9.1 Background 153

9.2 Main Results 161

9.3 Appendix 181

Bibliography 183

10 General Grüss and Ostrowski Inequalities Using s-Convexity 185

10.1 Background 185

10.2 Main Results 189

Bibliography 207

11 Essential and s-Convexity Ostrowski and Grüss Inequalities Using Several Functions 209

11.1 Introduction 209

11.2 Background 211

11.3 Main Results 211

Bibliography 235

12 General Fractional Hermitc-Hadamard Inequalities Using m-Convexity and (s, m)-Convexity 237

12.1 Background 237

12.2 Main Results 244

Bibliography 255

13 About the Reduction Method in Fractional Calculus and Fractional Ostrowski Inequalities 257

13.1 The Reduction Method in Fractional Calculus 257

13.2 Fractional Ostrowski Type Inequalities 266

13.3 Addendum 273

Bibliography 275

Index 277