Marek Kuczma An Introduction to the Theory of Functional Equations and Inequalities Cauchy s Equation and Jensen s Inequality Second Edition Edited by Attila Gilányi Birkhäuser Basel Boston Berlin
Editor: Attila Gilányi Institute of Mathematics University of Debrecen P.O. Box 12 4010 Debrecen Hungary e-mail: gilanyi@math.klte.hue 2000 Mathematical Subject Classification: 39B05, 39B22, 39B32, 39B52, 39B62, 39B82, 26A51, 26B25 The first edition was published in 1985 by Uniwersytet Slaski (Katowicach) (Silesian University of Katowice) and Pánstwowe Wydawnictwo Naukowe (Polish Scientific Publishers) Uniwersytet Slaski and Pánstwowe Wydawnictwo Naukowe Library of Congress Control Number: 2008939524 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. ISBN 978-3-7643-8748-8 Birkhäuser Verlag AG, Basel Boston Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. 2009 Birkhäuser Verlag AG Basel Boston Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced of chlorine-free pulp. TCF Printed in Germany ISBN 978-3-7643-8748-8 e-isbn 978-3-7643-8749-5 9 8 7 6 5 4 3 2 1 www.birkhauser.ch
Preface to the Second Edition The first edition of Marek Kuczma s book An Introduction to the Theory of Functional Equations and Inequalities was published more than 20 years ago. Since then it has been considered as one of the most important monographs on functional equations, inequalities and related topics. As János Aczél wrote in Mathematical Reviews... this is a very useful book and a primary reference not only for those working in functional equations, but mainly for those in other fields of mathematics and its applications who look for a result on the Cauchy equation and/or the Jensen inequality. Based on the considerably high demand for the book, which has even increased after the first edition was sold out several years ago, we have decided to prepare its second edition. It corresponds to the first one and keeps its structure and organization almost everywhere. The few changes which were made are always marked by footnotes. Several colleagues helped us in the preparation of the second edition. We cordially thank Roman Ger for his advice and help during the whole publication process, Karol Baron and Zoltán Boros for their conscientious proofreading, and Szabolcs Baják for typing and continuously correcting the manuscript. We are grateful to Eszter Gselmann, Fruzsina Mészáros, Gyöngyvér Péter and Pál Burai for typesetting several chapters, and we would like to thank the publisher, Birkhäuser, for undertaking and helping with the publication. The new edition of Marek Kuczma s book is paying tribute to the memory of the highly respected teacher, the excellent mathematician and one of the most outstanding researchers of functional equations and inequalities. Debrecen, October 2008 Attila Gilányi
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 Borel sets.................................. 25 2.4 The space z................................. 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
viii 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
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-Pólya majorization principle............. 211 8.5 Lim s inequality............................... 214 8.6 Hadamard inequality............................ 215 8.7 Petrović 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 A,B,C............................. 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 A, B, C 10.1 A Hahn-Banach theorem.......................... 257 10.2 The class B................................. 260 10.3 The class C................................. 266 10.4 The class A................................. 267 10.5 Set-theoretic operations.......................... 269 10.6 The classes D................................ 271 10.7 The classes A C and B 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 Erdős sets.................................. 288 11.6 Lusin sets.................................. 294 11.7 Perfect sets................................. 299 11.8 The operations R and U.......................... 301
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 Hosszú equation.............................. 374 13.8 Mikusiński 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
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
Introduction The present book is based on the course given by the author at the Silesian University in the academic year 1974/75, entitled Additive Functions and Convex Functions. Writing it, we have used excellent notes taken by Professor K. Baron. It may be objected whether an exposition devoted entirely to a single equation (Cauchy s Functional Equation) and a single inequality (Jensen s Inequality) deserves the name An introduction to the Theory of Functional Equations and Inequalities. However, the Cauchy equation plays such a prominent role in the theory of functional equations that the title seemed appropriate. Every adept of the theory of functional equations should be acquainted with the theory of the Cauchy equation. And a systematic exposition of the latter is still lacking in the mathematical literature, the results being scattered over particular papers and books. We hope that the present book will fill this gap. The properties of convex functions (i.e., functions fulfilling the Jensen inequality) resemble so closely those of additive functions (i.e., functions satisfying the Cauchy equation) that it seemed quite appropriate to speak about the two classes of functions together. Even in such a large book it was impossible to cover the whole material pertinent to the theory of the Cauchy equation and Jensen s inequality. The exercises at the end of each chapter and various bibliographical hints will help the reader to pursue further his studies of the subject if he feels interested in further developments of the theory. In the theory of convex functions we have concentrated ourselves rather on this part of the theory which does not require regularity assumptions about the functions considered. Continuous convex functions are only discussed very briefly in Chapter 7. The emphasis in the book lies on the theory. There are essentially no examples or applications. We hope that the importance and usefulness of convex functions and additive functions is clear to everybody and requires no advertising. However, many examples of applications of the Cauchy equation may be found, in particular, in books Aczél [5] and Dhombres [68]. Concerning convex functions, numerous examples are scattered throughout almost the whole literature on mathematical analysis, but especially the reader is referred to special books on convex functions quoted in 5.3. We have restricted ourselves to consider additive functions and convex functions defined in (the whole or subregions of) N-dimensional euclidean space R N.Thisgives the exposition greater uniformity. However, considerable parts of the theory presented
xiv Introduction can be extended to more general spaces (Banach spaces, topological linear spaces). Such an approach may be found in some other books (Dhombres [68], Roberts-Varberg [267]). Only occasionally we consider some functional equations on groups or related algebraic structures. We assume that the reader has a basic knowledge of the calculus, theory of Lebesgue s measure and integral, algebra, topology and set theory. However, for the convenience of the reader, in the first part of the book we present such fragments of those theories which are often left out from the university courses devoted to them. Also, some parts which are usually included in the university courses of these subjects are also very shortly treated here in order to fix the notation and terminology. In the notation we have tried to follow what is generally used in the mathematical literature 1. The cardinality of a set A is denoted by card A. Thewordcountable or denumerable refers to sets whose cardinality is exactly ℵ 0. The topological closure and interior of A are denoted by cl A and int A. Some special letters are used to denote particular sets of numbers. And so N denotes the set of positive integers, whereas Z denotes the set of all integers. Q stands for the set of all rational numbers, R for the set of all real numbers, and C for the set of all complex numbers. The letter N is reserved to denote the dimension of the underlying space. The end of every proof is marked by the sign. Other symbols are introduced in the text, and for the convenience of the reader they are gathered in an index at the end of the volume. The book is divided in chapters, every chapter is divided into sections. When referring to an earlier formula, we use a three digit notation: (X.Y.Z) means formula Z in section Y in Chapter X. The same rule applies also to the numbering of theorems and lemmas. When quoting a section, we use a two digit notation: X.Y means section Y in Chapter X. The same rule applies also to exercises at the end of each chapter. The book is also divided in three parts, but this fact has no reflection in the numeration. Many colleagues from Poland and abroad have helped us with bibliographical hints and otherwise. We do not endeavour to mention all their names, but nonetheless we would like to thank them sincerely at this place. But at least two names must be mentioned: Professor R. Ger, and above all, Professor K. Baron, whose help was especially substantial, and to whom our debt of gratitude is particularly great. We thank also the authorities of the Silesian University in Katowice, which agreed to publish this book. We hope that the mathematical community of the world will find it useful. Katowice, July 1979 Marek Kuczma 1 The notation in the second edition has been slightly changed. The following sentences are modified accordingly.