An introduction to the theory of functional equations and inequalities: Cauchy's equation and Jensen's inequality
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Weitere beteiligte Personen: | |
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Basel [u.a.]
Birkhäuser
2009
|
| Ausgabe: | 2. ed. |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016314677&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XIV, 595 S. |
| ISBN: | 3764387483 9783764387488 |
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| 245 | 1 | 0 | |a An introduction to the theory of functional equations and inequalities |b Cauchy's equation and Jensen's inequality |c Marek Kuczma |
| 250 | |a 2. ed. |b ed. by Attila Gilányi | ||
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2009 | |
| 300 | |a XIV, 595 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Additive functions | |
| 650 | 4 | |a Convex functions | |
| 650 | 4 | |a Functional equations | |
| 650 | 4 | |a Inequalities (Mathematics) | |
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Datensatz im Suchindex
| _version_ | 1819259152416899072 |
|---|---|
| adam_text | IMAGE 1
CONTENTS
INTRODUCTION XIII
PART I PRELIMINARIES
1 SET THEORY 1.1 AXIOMS OF SET THEORY 3
1.2 ORDERED SETS 5
1.3 ORDINAL NUMBERS 6
1.4 SETS OF ORDINAL NUMBERS 8
1.5 CARDINALITY OF ORDINAL NUMBERS 10
1.6 TRANSFINITE INDUCTION 12
1.7 THE ZERMELO THEOREM 14
1.8 LEMMA OF KURATOWSKI-ZORN 15
2 TOPOLOGY 2.1 CATEGORY 19
2.2 BAIRE PROPERTY 23
2.3 BOREI SETS 25
2.4 THE SPACE 3 28
2.5 ANALYTIC SETS 32
2.6 OPERATION A 35
2.7 THEOREM OF MARCZEWSKI 37
2.8 CANTOR-BENDIXSON THEOREM 39
2.9 THEOREM OF S. PICCARD 42
3 MEASURE THEORY 3.1 OUTER AND INNER MEASURE 47
3.2 LINEAR TRANSFORMS 54
3.3 SATURATED NON-MEASURABLE SETS 56
3.4 LUSIN SETS 59
3.5 OUTER DENSITY 61
3.6 SOME LEMMAS 63
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/986812838
DIGITALISIERT DURCH
IMAGE 2
V I II CONTENTS
3.7 THEOREM OF STEINHAUS 67
3.8 NON-MEASURABLE SETS 71
4 ALGEBRA 4.1 LINEAR INDEPENDENCE AND DEPENDENCE 75
4.2 BASES 78
4.3 HOMOMORPHISMS 83
4.4 CONES 87
4.5 GROUPS AND SEMIGROUPS 89
4.6 PARTITIONS OF GROUPS 95
4.7 RINGS AND FIELDS 98
4.8 ALGEBRAIC INDEPENDENCE AND DEPENDENCE 101
4.9 ALGEBRAIC AND TRANSCENDENTAL ELEMENTS 103
4.10 ALGEBRAIC BASES 105
4.11 SIMPLE EXTENSIONS OF FIELDS 106
4.12 ISOMORPHISM OF FIELDS AND RINGS 108
PART II CAUCHY S FUNCTIONAL EQUATION AND JENSEN S INEQUALITY
5 ADDITIVE FUNCTIONS AND CONVEX FUNCTIONS 5.1 CONVEX SETS 117
5.2 ADDITIVE FUNCTIONS 128
5.3 CONVEX FUNCTIONS 130
5.4 HOMOGENEITY FIELDS 137
5.5 ADDITIVE FUNCTIONS ON PRODUCT SPACES 138
5.6 ADDITIVE FUNCTIONS ON C 139
6 ELEMENTARY PROPERTIES OF CONVEX FUNCTIONS 6.1 CONVEX FUNCTIONS ON
RATIONAL LINES 143
6.2 LOCAL BOUNDEDNESS OF CONVEX FUNCTIONS 148
6.3 THE LOWER HULL OF A CONVEX FUNCTIONS 150
6.4 THEOREM OF BERNSTEIN-DOETSCH 155
7 CONTINUOUS CONVEX FUNCTIONS 7.1 THE BASIC THEOREM 161
7.2 COMPOSITIONS AND INVERSES 162
7.3 DIFFERENCES QUOTIENTS 164
7.4 DIFFERENTIATION 168
7.5 DIFFERENTIAL CONDITIONS OF CONVEXITY 171
7.6 FUNCTIONS OF SEVERAL VARIABLES 174
7.7 DERIVATIVES OF A FUNCTION 177
7.8 DERIVATIVES OF CONVEX FUNCTIONS 180
7.9 DIFFERENTIABILITY OF CONVEX FUNCTIONS 188
7.10 SEQUENCES OF CONVEX FUNCTIONS 192
IMAGE 3
CONTENTS IX
8 INEQUALITIES 8.1 JENSEN INEQUALITY 197
8.2 JENSEN-STEFFENSEN INEQUALITIES 201
8.3 INEQUALITIES FOR MEANS 208
8.4 HARDY-LITTLEWOOD-POLYA MAJORIZATION PRINCIPLE 211
8.5 LIM S INEQUALITY 214
8.6 HADAMARD INEQUALITY 215
8.7 PETROVIC INEQUALITY 217
8.8 MULHOLLAND S INEQUALITY 218
8.9 THE GENERAL INEQUALITY OF CONVEXITY 223
9 BOUNDEDNESS AND CONTINUITY OF CONVEX FUNCTIONS AND ADDITIVE FUNCTIONS
9.1 THE CLASSES 21,93,(2: 227
9.2 CONSERVATIVE OPERATIONS 229
9.3 SIMPLE CONDITIONS 231
9.4 MEASURABILITY OF CONVEX FUNCTIONS 241
9.5 PLANE CURVES 242
9.6 SKEW CURVES 244
9.7 BOUNDEDNESS BELOW 246
9.8 RESTRICTIONS OF CONVEX FUNCTIONS AND ADDITIVE FUNCTIONS 251
10 THE CLASSES 21, 23, * 10.1 A HAHN-BANACH THEOREM 257
10.2 THE CLASS 93 260
10.3 THE CLASS 266
10.4 THE CLASS 21 267
10.5 SET-THEORETIC OPERATIONS 269
10.6 THE CLASSES 2 271
10.7 THE CLASSES 2L C AND 23 C 276
11 PROPERTIES OF HAMEL BASES 11.1 GENERAL PROPERTIES 281
11.2 MEASURE 282
11.3 TOPOLOGICAL PROPERTIES 285
11.4 BURSTIN BASES 285
11.5 ERDOES SETS 288
11.6 LUSIN SETS 294
11.7 PERFECT SETS 299
11.8 THE OPERATIONS R AND U 301
IMAGE 4
X CONTENTS
12 FURTHER PROPERTIES OF ADDITIVE FUNCTIONS AND CONVEX FUNCTIONS 12.1
GRAPHS 305
12.2 ADDITIVE FUNCTIONS 308
12.3 CONVEX FUNCTIONS 313
12.4 BIG GRAPH 316
12.5 INVERTIBLE ADDITIVE FUNCTIONS 322
12.6 LEVEL SETS 327
12.7 PARTITIONS 330
12.8 MONOTONICITY 335
PART III RELATED TOPICS
13 RELATED EQUATIONS 13.1 THE REMAINING CAUCHY EQUATIONS 343
13.2 JENSEN EQUATION 351
13.3 PEXIDER EQUATIONS 355
13.4 MULTIADDITIVE FUNCTIONS 363
13.5 CAUCHY EQUATION ON AN INTERVAL 367
13.6 THE RESTRICTED CAUCHY EQUATION 369
13.7 HOSSZU EQUATION 374
13.8 MIKUSINSKI EQUATION 376
13.9 AN ALTERNATIVE EQUATION 380
13.10THE GENERAL LINEAR EQUATION 382
14 DERIVATIONS AND AUTOMORPHISMS 14.1 DERIVATIONS 391
14.2 EXTENSIONS OF DERIVATIONS 394
14.3 RELATIONS BETWEEN ADDITIVE FUNCTIONS 399
14.4 AUTOMORPHISMS OF R 402
14.5 AUTOMORPHISMS OF C 403
14.6 NON-TRIVIAL ENDOMORPHISMS OF C 406
15 CONVEX FUNCTIONS OF HIGHER ORDERS 15.1 THE DIFFERENCE OPERATOR 415
15.2 DIVIDED DIFFERENCES 421
15.3 CONVEX FUNCTIONS OF HIGHER ORDER 429
15.4 LOCAL BOUNDEDNESS OF P-CONVEX FUNCTIONS 432
15.5 OPERATION H 435
15.6 CONTINUOUS P-CONVEX FUNCTIONS 439
15.7 CONTINUOUS P-CONVEX FUNCTIONS. CASE N = 1 442
15.8 DIFFERENTIABILITY OF P-CONVEX FUNCTIONS 444
15.9 POLYNOMIAL FUNCTIONS 446
IMAGE 5
CONTENTS XI
16 SUBADDITIVE FUNCTIONS 16.1 GENERAL PROPERTIES 455
16.2 BOUNDEDNESS. CONTINUITY 458
16.3 DIFFERENTIABILITY 465
16.4 SUBLINEAR FUNCTIONS 471
16.5 NORM 473
16.6 INFINITARY SUBADDITIVE FUNCTIONS 475
17 NEARLY ADDITIVE FUNCTIONS AND NEARLY CONVEX FUNCTIONS 17.1
APPROXIMATELY ADDITIVE FUNCTIONS 483
17.2 APPROXIMATELY MULTIADDITIVE FUNCTIONS 485
17.3 FUNCTIONS WITH BOUNDED DIFFERENCES 486
17.4 APPROXIMATELY CONVEX FUNCTIONS 490
17.5 SET IDEALS 498
17.6 ALMOST ADDITIVE FUNCTIONS 505
17.7 ALMOST POLYNOMIAL FUNCTIONS 510
17.8 ALMOST CONVEX FUNCTIONS 515
17.9 ALMOST SUBADDITIVE FUNCTIONS 524
18 EXTENSIONS OF HOMOMORPHISMS 18.1 COMMUTATIVE DIVISIBLE GROUPS 535
18.2 THE SIMPLEST CASE OF S GENERATING X 537
18.3 A GENERALIZATION 540
18.4 FURTHER EXTENSION THEOREMS 546
18.5 CAUCHY EQUATION ON A CYLINDER 551
18.6 CAUCHY NUCLEUS 556
18.7 THEOREM OF GER 560
18.8 INVERSE ADDITIVE FUNCTIONS 564
18.9 CONCLUDING REMARKS 569
BIBLIOGRAPHY 571
INDICES
INDEX OF SYMBOLS 587
SUBJECT INDEX 589
INDEX OF NAMES 593
|
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| author | Kuczma, Marek 1935-1991 |
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| dewey-raw | 515.75 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV023112102 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T13:08:55Z |
| institution | BVB |
| isbn | 3764387483 9783764387488 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016314677 |
| oclc_num | 214309410 |
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| owner_facet | DE-20 DE-703 DE-824 DE-11 DE-83 |
| physical | XIV, 595 S. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Birkhäuser |
| record_format | marc |
| spellingShingle | Kuczma, Marek 1935-1991 An introduction to the theory of functional equations and inequalities Cauchy's equation and Jensen's inequality Additive functions Convex functions Functional equations Inequalities (Mathematics) Funktionalgleichung (DE-588)4018923-5 gnd Ungleichung (DE-588)4139098-2 gnd |
| subject_GND | (DE-588)4018923-5 (DE-588)4139098-2 |
| title | An introduction to the theory of functional equations and inequalities Cauchy's equation and Jensen's inequality |
| title_auth | An introduction to the theory of functional equations and inequalities Cauchy's equation and Jensen's inequality |
| title_exact_search | An introduction to the theory of functional equations and inequalities Cauchy's equation and Jensen's inequality |
| title_full | An introduction to the theory of functional equations and inequalities Cauchy's equation and Jensen's inequality Marek Kuczma |
| title_fullStr | An introduction to the theory of functional equations and inequalities Cauchy's equation and Jensen's inequality Marek Kuczma |
| title_full_unstemmed | An introduction to the theory of functional equations and inequalities Cauchy's equation and Jensen's inequality Marek Kuczma |
| title_short | An introduction to the theory of functional equations and inequalities |
| title_sort | an introduction to the theory of functional equations and inequalities cauchy s equation and jensen s inequality |
| title_sub | Cauchy's equation and Jensen's inequality |
| topic | Additive functions Convex functions Functional equations Inequalities (Mathematics) Funktionalgleichung (DE-588)4018923-5 gnd Ungleichung (DE-588)4139098-2 gnd |
| topic_facet | Additive functions Convex functions Functional equations Inequalities (Mathematics) Funktionalgleichung Ungleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016314677&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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