An introduction to the theory of functional equations and inequalities

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

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

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Editor:

Attila Gilányi

Institute of Mathematics

University of Debrecen

P.O. Box 12

4010 Debrecen

Hungary

e-mail: [email protected]

´

´

Preface to the Second Edition

The first edition of Marek Kuczma’s book An Introduction to the Theory of Func￾tional 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 equa￾tions, inequalities and related topics. As J´anos Acz´el 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 appli￾cations 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 cor￾dially thank Roman Ger for his advice and help during the whole publication process,

Karol Baron and Zolt´an Boros for their conscientious proofreading, and Szabolcs

Baj´ak for typing and continuously correcting the manuscript. We are grateful to

Eszter Gselmann, Fruzsina M´esz´aros, Gy¨ongyv´er P´eter and P´al Burai for typesetting

several chapters, and we would like to thank the publisher, Birkh¨auser, for undertak￾ing 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 Gil´anyi

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-P´olya majorization principle . . . . . . . . . . . . . 211

8.5 Lim’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

8.6 Hadamard inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

8.7 Petrovi´c 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 AC and BC . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Erd˝os 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 Hossz´u equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

13.8 Mikusi´nski 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 sys￾tematic 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 Acz´el [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 RN . This gives

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

literature1. The cardinality of a set A is denoted by cardA. The word countable or

denumerable refers to sets whose cardinality is exactly ℵ0. 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

1The notation in the second edition has been slightly changed. The following sentences are modified

accordingly.

Part I

Preliminaries