Handbook of means and their inequalities / by P.S. Bullen.

Format:

Book

Author/Creator:

Bullen, P. S., 1928-

Contributor:

Rosengarten Family Fund.

Series:

Mathematics and its applications (Kluwer Academic Publishers) ; v. 560.

Mathematics and its applications ; v. 560

Language:

English

Subjects (All):

Inequalities (Mathematics).

Physical Description:

xxvii, 537 pages : illustrations ; 25 cm.

Place of Publication:

Dordrecht ; Boston : Kluwer Academic Publishers, [2003]

Contents:

1. Referencing xix

2. Bibliographic References xix

3. Symbols for Some Important Inequalities xix

4. Numbers, Sets and Set Functions xx

5. Intervals xx

6. n-tuples xxi

7. Matrices xxii

8. Functions xxii

9. Various xxiii

1. Properties of Polynomials 1

1.1 Some Basic Results 1

1.2 Some Special Polynomials 3

2. Elementary Inequalities 4

2.1 Bernoulli's Inequality 4

2.2 Inequalities Involving Some Elementary Functions 6

3. Properties of Sequences 11

3.1 Convexity and Bounded Variation of Sequences 11

3.2 Log-convexity of Sequences 16

3.3 An Order Relation for Sequences 21

4. Convex Functions 25

4.1 Convex Functions of Single Variable 25

4.2 Jensen's Inequality 30

4.3 The Jensen-Steffensen Inequality 37

4.4 Reverse and Converse Jensen Inequalities 43

4.5 Other Forms of Convexity 48

4.6 Convex Functions of Several Variables 50

4.7 Higher Order Convexity 54

4.8 Schur Convexity 57

4.9 Matrix Convexity 58

Chapter II The Arithmetic, Geometric and Harmonic Means 60

1. Definitions and Simple Properties 60

1.1 The Arithmetic Mean 60

1.2 The Geometric and Harmonic Means 64

1.3 Some Interpretations and Applications 66

2. The Geometric Mean-Arithmetic Mean Inequality 71

2.1 Statement of the Theorem 71

2.2 Some Preliminary Results 72

2.3 Some Geometrical Interpretations 82

2.4 Proofs of the Geometric Mean-Arithmetic Mean Inequality 84

2.5 Applications of the Geometric Mean-Arithmetic Mean Inequality 119

3. Refinements of the Geometric Mean-Arithmetic Mean Inequality 125

3.1 The Inequalities of Rado and Popoviciu 125

3.2 Extensions of the Inequalities of Rado and Popoviciu 129

3.3 A Limit Theorem of Everitt 133

3.4 Nanjundiah's Inequalities 136

3.5 Kober-Diananda Inequalities 141

3.6 Redheffer's Recurrent Inequalities 145

3.7 The Geometric Mean-Arithmetic Mean Inequality with General Weights 148

3.8 Other Refinements of Geometric Mean-Arithmetic Mean Inequality 149

4. Converse Inequalities 154

4.1 Bounds for the Differences of the Means 154

4.2 Bounds for the Ratios of the Means 157

5. Some Miscellaneous Results 160

5.1 An Inductive Definition of the Arithmetic Mean 160

5.2 An Invariance Property 160

5.3 Cebisev's Inequality 161

5.4 A Result of Diananda 165

5.5 Intercalated Means 166

5.6 Zeros of a Polynomial and Its Derivative 170

5.7 Nanson's Inequality 170

5.8 The Pseudo Arithmetic Means and Pseudo Geometric Means 171

5.9 An Inequality Due to Mercer 174

Chapter III The Power Means 175

1. Definitions and Simple Properties 175

2. Sums of Powers 178

2.1 Holder's Inequality 178

2.2 Cauchy's Inequality 183

2.3 Power sums 185

2.4 Minkowski's Inequality 189

2.5 Refinements of the Holder, Cauchy and Minkowski Inequalities 192

3. Inequalities Between the Power Means 202

3.1 The Power Mean Inequality 202

3.2 Refinements of the Power Mean Inequality 216

4. Converse Inequalities 229

4.1 Ratios of Power Means 230

4.2 Differences of Power Means 238

4.3 Converses of the Cauchy, Holder and Minkowski Inequalities 240

5. Other Means Defined Using Powers 245

5.1 Counter-Harmonic Means 245

5.2 Generalizations of the Counter-Harmonic Means 248

5.3 Mixed Means 253

6. Some Other Results 256

6.1 Means on the Move 256

6.2 Hlawka-type inequalities 258

6.3 p-Mean Convexity 260

6.4 Various Results 260

Chapter IV Quasi-Arithmetic Means 266

1. Definitions and Basic Properties 266

1.1 The Definition and Examples 266

1.2 Equivalent Quasi-arithmetic Means 271

2. Comparable Means and Functions 273

3. Results of Rado-Popoviciu Type 280

3.1 Some General Inequalities 280

3.2 Some Applications of the General Inequalities 282

4 Further Inequalities 285

4.1 Cakalov's Inequality 286

4.2 A Theorem of Godunova 288

4.3 A Problem of Oppenheim 290

4.4 Ky Fan's Inequality 294

4.5 Means on the Move 298

5. Generalizations of the Holder and Minkowski Inequalities 299

6. Converse Inequalities 307

7. Generalizations of the Quasi-arithmetic Means 310

7.1 A Mean of Bajraktarevic 310

7.2 Further Results 316

Chapter V Symmetric Polynomial Means 321

1. Elementary Symmetric Polynomials and Their Means 321

2. The Fundamental Inequalities 324

3. Extensions of S(r;s) of Rado-Popoviciu Type 334

4. The Inequalities of Marcus & Lopes 338

5. Complete Symmetric Polynomial Means; Whiteley Means 341

5.1 The Complete Symmetric Polynomial Means 341

5.2 The Whiteley Means 343

5.3 Some Forms of Whiteley 349

5.4 Elementary Symmetric Polynomial Means as Mixed Means 356

6. The Muirhead Means 357

7. Further Generalizations 364

7.1 The Hamy Means 364

7.2 The Hayashi Means 365

7.3 The Biplanar Means 366

7.4 The Hypergeometric Mean 366

Chapter VI Other Topics 368

1. Integral Means and Their Inequalities 368

1.1 Generalities 368

1.2 Basic Theorems 370

1.3 Further Results 377

2. Two Variable Means 384

2.1 The Generalized Logarithmic and Extended Means 385

2.2 Mean Value Means 403

2.3 Means and Graphs 406

2.4 Taylor Remainder Means 409

2.5 Decomposition of Means 412

3. Compounding of Means 413

3.1 Compound means 413

3.2 The Arithmetico-geometric Mean and Variants 417

4. Some General Approaches to Means 420

4.1 Level Surface Means 420

4.2 Corresponding Means 422

4.3 A Mean of Galvani 423

4.4 Admissible Means of Bauer 423

4.5 Segre Functions 425

4.6 Entropic Means 427

5. Mean Inequalities for Matrices 429

6. Axiomatization of Means 435.

Notes:

Includes bibliographical references (pages 439-509) and indexes.

Local Notes:

Acquired for the Penn Libraries with assistance from the Rosengarten Family Fund.

ISBN:

1402015224

OCLC:

52695402