Recent advances in geometric inequalities:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht u.a.
Kluwer
1989
|
| Series: | Mathematics and its applications / East European series
28 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 710 S. |
| ISBN: | 9027725659 |
Staff View
MARC
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| 100 | 1 | |a Mitrinović, Dragoslav S. |d 1908-1995 |e Verfasser |0 (DE-588)128427183 |4 aut | |
| 245 | 1 | 0 | |a Recent advances in geometric inequalities |c D. S. Mitrinović ; J. E. Pečarić ; V. Volenec |
| 264 | 1 | |a Dordrecht u.a. |b Kluwer |c 1989 | |
| 300 | |a XIX, 710 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications / East European series |v 28 | |
| 650 | 4 | |a Geometrie - Ungleichung | |
| 650 | 0 | 7 | |a Geometrische Ungleichung |0 (DE-588)4705164-4 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Geometrische Ungleichung |0 (DE-588)4705164-4 |D s |
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| 700 | 1 | |a Pečarić, Josip E. |d 1948- |e Verfasser |0 (DE-588)135806941 |4 aut | |
| 700 | 1 | |a Volenec, Vladimir |e Verfasser |4 aut | |
| 810 | 2 | |a East European series |t Mathematics and its applications |v 28 |w (DE-604)BV000006709 |9 28 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001598106&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
Record in the Search Index
| _version_ | 1805073274504216576 |
|---|---|
| adam_text |
Contents
SERES EDITOR'S PREFACE xi
PREFACE xiii
NOTATION AND ABBREVIATIONS xvii
ORGANIZATION OF THE BOOK xix
CHAPTER I. THE EXISTENCE OF A TRIANGLE 1
0. Introduction 1
1. The Fundamental Inequality 1
1.1. History 1
1.2. Proofs 5
1.3. A Geometric Interpretation 9
2. The Existence of a Triangle with Given Elements 11
3. Some Other Results 18
CHAPTER II. DUALITY BETWEEN GEOMETRIC INEQUALITIES
AND INEQUALITIES FOR POSITIVE NUMBERS 26
0. Introduction 26
1. Geometry of the Duality (a, b, c,) = (x, y, z,) 26
2. Transformations 27
3. Examples 30
4. Some Important Non Negative Quadratic Forms 33
CHAPTER III. HOMOGENEOUS SYMMETRIC POLYNOMIAL
GEOMETRIC INEQUALITIES 37
0. Introduction 37
1. General Results of P. J. van Albada and K. B. Stolarsky 37
2. Special Inequalities 39
3. Best Possible Inequalities 42
4. Generalization of Gerretsen's Inequalities 45
CHAPTER IV. DUALITY BETWEEN DIFFERENT TRIANGLE
INEQUALITIES AND TRIANGLE INEQUALITIES
WITH (R, r, s) 49
1. Some General Considerations and Some Applications 49
2. Some Equivalent Forms 52
V
vi CONTENTS
CHAPTER V. TRANSFORMATIONS FOR THE ANGLES OF A
TRIANGLE 64
1. Some Applications of Pexider's Functional Equation in
Geometry 64
2. Some Applications 70
3. On the Obtaining of Analytic Inequalities from Geometric
Ones 71
CHAPTER VI. SOME TRIGONOMETRIC INEQUALITIES 74
0. Introduction 74
1. Asymmetric Trigonometric Inequalities 74
2. Some Trigonometric Identities 85
3. Some Applications 90
4. Open Questions 98
CHAPTER VII. SOME OTHER TRANSFORMATIONS 101
0. Introduction 101
1. Square Root Transformation 102
2. Generalizations of the Finsler Hadwiger Inequality and
Applications 104
3. Some Trigonometric Transformations 106
4. The Median Dual Transformation and Its Generalizations 109
5. Transformations Tt, Te2 and T# i 112
6. Parallelogram Transformations 121
7. A Transformation for Acute Triangles 117
CHAPTER VIII. CONVEX FUNCTIONS AND GEOMETRIC
INEQUALITIES 121
1. Jensen's and Related Inequalities and Geometric Inequalities 121
2. Majorization and Geometric Inequalities and Geometric
Inequalities 121
2.1. Inequalities for the Sides of a Triangle and Polygon 129
2.2. Inequalities for the Angles of a Triangle and Polygon 133
2.3. Majorization and Other Elements of a Triangle 138
2.4. Majorization and Isoperimetric Type Inequalities 139
3. Applications of Some Other Inequalities for Convex Functions 140
CHAPTER IX. MISCELLANEOUS INEQUALITIES WITH ELE¬
MENTS OF A TRIANGLE 144
1. Inequalities Involving only the Sides of a Triangle 144
CONTENTS vii
2. Inequalities for the Angles of a Triangle 149
2.1. Bager's Graphs 149
2.2. Miscellaneous Inequalities for the Angles of a Triangle 154
3. Inequalities with (R, r, s) 165
4. Inequalities for the Sides and the Angles of a Triangle 168
5. Inequalities with (a, b, c) and (R, r, s or F) 170
6. Inequalities Involving A, B, C and R, r, s or F 180
7. Inequalities with (a, b, c), (A, B, Q and (R, r.sorF) 190
8. Inequalities for the Radii of Excircles and Other Elements of a
Triangle 192
9. Inequalities Involving Altitudes and Other Elements of a
Triangle 200
10. Inequalities with Medians and Other Elements of a Triangle 209
11. Inequalities with Angle Bisectors and Other Elements of a
Triangle 217
12. Inequalities with Some Elements of a Triangle Extended to the
Circumcircle 229
CHAPTER X. SPECIAL TRIANGLES 232
1. On the Triangles Satisfying a2 + b2 + c2 | kR2 232
2. On Some Inequalities for Acute (Non Obtuse) Triangles 240
2.0. Introduction 240
2.1. A Question of Ono 240
2.2. An Inequality of Gridasow 242
2.3. An Inequality of Walker 247
3. On Some Inequalities for Obtuse (Non acute) Triangles 251
3.1. Emmerich's Inequality 251
3.2. Inequalities of Bottema and Groenman 253
4. Some Other Classes of Special Triangles 255
4.0. Introduction 255
4.1. Inequalities for Triangles of Bager's Type I or II 256
4.2. Inequalities of S.G.Guba for Special Triangles 261
4.3. Inequalities of I. Paasche for Special Triangles 263
4.4. Some Further Remarks 265
5. Some Other Results for Special Triangles 267
CHAPTER XI. TRIANGLE AND POINT 278
1. Some Applications of a Generalization of the Leibniz Identity 278
2. Some Transformations 293
2.1. Inversion 293
2.2. Reciprocation 295
2.3. Isogonal Conjugates 295
viii CONTENTS
.2.4. Pedal Triangle 296
3. Some Further Remarks on the Polar Moment of Inertia In¬
equality 296
4. Two Simple Methods for Generating Inequalities Involving
Triangle and Point 301
4.1. Applications of Jensen's Inequality for Convex Functions
and of the Inequality for Means 301
4.2. Triangle Inequalities from the Triangle Inequality 309
5. ErdOs Mordell's Related Inequalities 313
6. Miscellaneous Inequalities Involving Characteristic Points 319
7. Miscellaneous Inequalities Involving Arbitrary Points 333
CHAPTER XII. INEQUALITIES WITH SEVERAL TRIANGLES 340
1. Inequalities Related to Two Triangles Inscribed One in the
Other 340
2. Some Other Inequalities with Triangles Connected to the Given
Triangle 347
3. The Neuberg Pedoe and the Oppenheim Inequalities 354
3.0. Introduction 354
3.1. The Neuberg Pedoe and the Oppenheim Inequality 355
3.2. Comment by O. Bottema: On the Mixed Area of Two
Triangles 357
3.3. On a Result of Carlitz 358
3.4. Sharpening the Neuberg Pedoe and the Oppenheim
Inequality 359
3.5. Further Generalizations of the Neuberg Pedoe Inequality 364
3.6. Further Generalizations of the Oppenheim Inequality 366
4. On O. Bottema's Inequality for Two Triangles 371
5. Miscellaneous Inequalities Involving Elements of Several
Triangles 375
CHAPTER XIII. THE MOBIUS NEUBERG AND THE MOBIUS
POMPEIU THEOREMS 385
CHAPTER XIV. INEQUALITIES FOR QUADRILATERALS 401
CHAPTER XV. INEQUALITIES FOR POLYGONS 412
CHAPTER XVI. INEQUALITIES FOR A CIRCLE 433
CHAPTER XVII. PARTICULAR INEQUALITIES IN PLANE
GEOMETRY 443
CONTENTS ix
1. Some Isoperimetric Inequalities 443
2. Various Particular Inequalities 448
CHAPTERXVIII. INEQUALITIES FOR SIMPLEXES INEn(n 2) 463
1. Inequalities for r, p;, hl Fi (i = 1, 2, ., n + 1) in a Simplex 463
2. Inequalities for the Simplex and a Point 473
3. Other Inequalities for a Simplex 515
4. Inequalities for Two (or More) Simplexes 533
CHAPTER XIX. INEQUALITIES FOR TETRAHEDRA 545
1. Inequalities for the Edge Lengths and Face Areas of a
Tetrahedron 545
2. Inequalities for the Volume, the Circumradius and Other
Elements of a Tetrahedron 553
3. Other Inequalities for Tetrahedra 560
4. Inequalities for Special Tetrahedra 567
CHAPTERXX. OTHER INEQUALITIES INEn{n 2) 576
1. Inequalities for Convex Polyhedra 576
2. Inequalities for Prisms 592
3. Inequalities for Pyramids 602
4. Inequalities for the Regular Convex Polyhedra and Polyhedra
Isomorphic to Them 620
5. Inequalities for Quadric Surfaces 626
6. Inequalities for Spherical Triangles 636
7. Other Inequalities in E? 641
8. Inequalities for Convex Polytopes in (£" (ji 2) 653
9. Other Inequalities in E" (n 2) 659
APPENDIX 677
1. Comments, Additions and Corrections for GI 677
2. Supplement to Chapters I XX 682
NAME INDEX 695
SUBJECT INDEX 706 |
| any_adam_object | 1 |
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| spelling | Mitrinović, Dragoslav S. 1908-1995 Verfasser (DE-588)128427183 aut Recent advances in geometric inequalities D. S. Mitrinović ; J. E. Pečarić ; V. Volenec Dordrecht u.a. Kluwer 1989 XIX, 710 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications / East European series 28 Geometrie - Ungleichung Geometrische Ungleichung (DE-588)4705164-4 gnd rswk-swf Geometrische Ungleichung (DE-588)4705164-4 s DE-604 Pečarić, Josip E. 1948- Verfasser (DE-588)135806941 aut Volenec, Vladimir Verfasser aut East European series Mathematics and its applications 28 (DE-604)BV000006709 28 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001598106&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mitrinović, Dragoslav S. 1908-1995 Pečarić, Josip E. 1948- Volenec, Vladimir Recent advances in geometric inequalities Geometrie - Ungleichung Geometrische Ungleichung (DE-588)4705164-4 gnd |
| subject_GND | (DE-588)4705164-4 |
| title | Recent advances in geometric inequalities |
| title_auth | Recent advances in geometric inequalities |
| title_exact_search | Recent advances in geometric inequalities |
| title_full | Recent advances in geometric inequalities D. S. Mitrinović ; J. E. Pečarić ; V. Volenec |
| title_fullStr | Recent advances in geometric inequalities D. S. Mitrinović ; J. E. Pečarić ; V. Volenec |
| title_full_unstemmed | Recent advances in geometric inequalities D. S. Mitrinović ; J. E. Pečarić ; V. Volenec |
| title_short | Recent advances in geometric inequalities |
| title_sort | recent advances in geometric inequalities |
| topic | Geometrie - Ungleichung Geometrische Ungleichung (DE-588)4705164-4 gnd |
| topic_facet | Geometrie - Ungleichung Geometrische Ungleichung |
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