An Introduction to the Theory of Functional Equations and Inequalities

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2 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

3 Editor: Attila Gilányi Institute of Mathematics University of Debrecen P.O. Box Debrecen Hungary 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: 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 < ISBN 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 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 e-isbn

4 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

5 Contents Introduction xiii Part I Preliminaries 1 Set Theory 1.1 Axioms of Set Theory Ordered sets Ordinal numbers Sets of ordinal numbers Cardinality of ordinal numbers Transfinite induction The Zermelo theorem Lemma of Kuratowski-Zorn Topology 2.1 Category Baire property Borel sets The space z Analytic sets Operation A Theorem of Marczewski Cantor-Bendixson theorem Theorem of S. Piccard Measure Theory 3.1 Outer and inner measure Linear transforms Saturated non-measurable sets Lusin sets Outer density Some lemmas

6 viii Contents 3.7 Theorem of Steinhaus Non-measurable sets Algebra 4.1 Linear independence and dependence Bases Homomorphisms Cones Groups and semigroups Partitions of groups Rings and fields Algebraic independence and dependence Algebraic and transcendental elements Algebraic bases Simple extensions of fields Isomorphism of fields and rings Part II Cauchy s Functional Equation and Jensen s Inequality 5 Additive Functions and Convex Functions 5.1 Convex sets Additive functions Convex functions Homogeneity fields Additive functions on product spaces Additive functions on C Elementary Properties of Convex Functions 6.1 Convex functions on rational lines Local boundedness of convex functions The lower hull of a convex functions Theorem of Bernstein-Doetsch Continuous Convex Functions 7.1 The basic theorem Compositions and inverses Differences quotients Differentiation Differential conditions of convexity Functions of several variables Derivatives of a function Derivatives of convex functions Differentiability of convex functions Sequences of convex functions

7 Contents ix 8 Inequalities 8.1 Jensen inequality Jensen-Steffensen inequalities Inequalities for means Hardy-Littlewood-Pólya majorization principle Lim s inequality Hadamard inequality Petrović inequality Mulholland s inequality The general inequality of convexity Boundedness and Continuity of Convex Functions and Additive Functions 9.1 The classes A,B,C Conservative operations Simple conditions Measurability of convex functions Plane curves Skew curves Boundedness below Restrictions of convex functions and additive functions The Classes A, B, C 10.1 A Hahn-Banach theorem The class B The class C The class A Set-theoretic operations The classes D The classes A C and B C Properties of Hamel Bases 11.1 General properties Measure Topological properties Burstin bases Erdős sets Lusin sets Perfect sets The operations R and U

8 x Contents 12 Further Properties of Additive Functions and Convex Functions 12.1 Graphs Additive functions Convex functions Big graph Invertible additive functions Level sets Partitions Monotonicity Part III Related Topics 13 Related Equations 13.1 The remaining Cauchy equations Jensen equation Pexider equations Multiadditive functions Cauchy equation on an interval The restricted Cauchy equation Hosszú equation Mikusiński equation An alternative equation The general linear equation Derivations and Automorphisms 14.1 Derivations Extensions of derivations Relations between additive functions Automorphisms of R Automorphisms of C Non-trivial endomorphisms of C Convex Functions of Higher Orders 15.1 The difference operator Divided differences Convex functions of higher order Local boundedness of p-convex functions Operation H Continuous p-convex functions Continuous p-convex functions. Case N = Differentiability of p-convex functions Polynomial functions

9 Contents xi 16 Subadditive Functions 16.1 General properties Boundedness. Continuity Differentiability Sublinear functions Norm Infinitary subadditive functions Nearly Additive Functions and Nearly Convex Functions 17.1 Approximately additive functions Approximately multiadditive functions Functions with bounded differences Approximately convex functions Set ideals Almost additive functions Almost polynomial functions Almost convex functions Almost subadditive functions Extensions of Homomorphisms 18.1 Commutative divisible groups The simplest case of S generating X A generalization Further extension theorems Cauchy equation on a cylinder Cauchy nucleus Theorem of Ger Inverse additive functions Concluding remarks Bibliography Indices Index of Symbols Subject Index Index of Names

10 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

11 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.