Advanced Inequalities

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World Scientific, 2011 - Mathematics - 410 pages
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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 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
Index
409
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