## Advanced InequalitiesThis 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 HardyOpial type inequalities are examined. Works on ordinary and distributional Taylor formulae with estimates for their remainders and applications as well as ChebyshevGruss, 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. |

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### Contents

1 Introduction | 1 |

2 Advanced Univariate Ostrowski Type Inequalities | 5 |

3 Higher Order Ostrowski Inequalities | 21 |

4 Multidimensional Euler Identity and Optimal Multidimensional Ostrowski Inequalities | 27 |

5 More on Multidimensional Ostrowski Type Inequalities | 81 |

6 Ostrowski Inequalities on Euclidean Domains | 93 |

7 High Order Ostrowski Inequalities on Euclidean Domains | 99 |

8 Ostrowski Inequalities on Spherical Shells | 109 |

18 Poincare and Sobolev Like Inequalities for Widder Derivatives | 215 |

19 Poincare and Sobolev Like Inequalities for Vector Valued Functions | 229 |

20 Poincare Type Inequalities for Semigroups Cosine and Sine Operator Functions | 243 |

21 HardyOpial Type Inequalities | 261 |

22 A Basic Sharp Integral Inequality | 271 |

23 Estimates of the Remainder in Taylors Formula | 275 |

24 The Distributional Taylor Formula | 293 |

25 ChebyshevGruss Type Inequalities Using Euler Type and Fink Identities | 305 |

9 Ostrowski Inequalities on Balls and Shells Via TaylorWidder Formula | 125 |

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

11 General Multivariate Weighted Opial Inequalities | 149 |

12 Opial Inequalities for Widder Derivatives | 161 |

13 Opial Inequalities for Linear Differential Operators | 171 |

14 Opial Inequalities for Vector Valued Functions | 179 |

15 Opial Inequalities for Semigroups | 187 |

16 Opial Inequalities for Cosine and Sine Operator Functions | 197 |

17 Poincare Like Inequalities for Linear Differential Operators | 209 |

26 Gruss Type Multivariate Integral Inequalities | 319 |

27 ChebyshevGruss Type Inequalities on Spherical Shells and Balls | 331 |

28 Multivariate ChebyshevGruss and Comparison of Integral Means Inequalities | 341 |

29 Multivariate Fink Type Identity Applied to Multivariate Inequalities | 365 |

Bibliography | 395 |

List of Symbols | 407 |

409 | |