An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality 2nd ed. 2009 edition

Marc Notes: Includes bibliographical references and indexes. Table of Contents: Introduction -- Part I. Preliminaries -- 1. Set Theory -- 1.1. Axioms of Set Theory -- 1.2. Ordered sets -- 1.3. Ordinal numbers -- 1.4. Sets of ordinal numbers -- 1.5. Cardinality of ordinal numbers -- 1.6. Transfinite induction -- 1.7. The Zermelo theorem -- 1.8. Lemma of Kuratowski-Zorn -- 2. Topology -- 2.1. Category -- 2.2. Baire property -- 2.3. Borel sets -- 2.4. The space [characters not reproducible] -- 2.5. Analytic sets -- 2.6. Operation A -- 2.7. Theorem of Marczewski -- 2.8. Cantor-Bendixson theorem -- 2.9. Theorem of S. Piccard -- 3. Measure Theory -- 3.1. Outer and inner measure -- 3.2. Linear transforms -- 3.3. Saturated non-measurable sets -- 3.4. Lusin sets -- 3.5. Outer density -- 3.6. Some lemmas -- 3.7. Theorem of Steinhaus -- 3.8. Non-measurable sets -- 4. Algebra -- 4.1. Linear independence and dependence -- 4.2. Bases -- 4.3. Homomorphisms -- 4.4. Cones -- 4.5. Groups and semigroups -- 4.6. Partitions of groups -- 4.7. Rings and fields -- 4.8. Algebraic independence and dependence -- 4.9. Algebraic and transcendental elements -- 4.10. Algebraic bases -- 4.11. Simple extensions of fields -- 4.12. 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 -- 5.2. Additive functions -- 5.3. Convex functions -- 5.4. Homogeneity fields -- 5.5. Additive functions on product spaces -- 5.6. Additive functions on C -- 6. Elementary Properties of Convex Functions -- 6.1. Convex functions on rational lines -- 6.2. Local boundedness of convex functions -- 6.3. The lower hull of a convex functions -- 6.4. Theorem of Bernstein-Doetsch -- 7. Continuous Convex Functions -- 7.1. The basic theorem -- 7.2. Compositions and inverses -- 7.3. Differences quotients -- 7.4. Differentiation -- 7.5. Differential conditions of convexity -- 7.6. Functions of several variables -- 7.7. Derivatives of a function -- 7.8. Derivatives of convex functions -- 7.9. Differentiability of convex functions -- 7.10. Sequences of convex functions -- 8. Inequalities -- 8.1. Jensen inequality -- 8.2. Jensen-Steffensen inequalities -- 8.3. Inequalities for means -- 8.4. Hardy-Littlewood-Polya majorization principle -- 8.5. Lim's inequality -- 8.6. Hadamard inequality -- 8.7. Petrovic inequality -- 8.8. Mulholland's inequality -- 8.9. The general inequality of convexity -- 9. Boundedness and Continuity of Convex Functions and Additive Functions -- 9.1. The classes U, B, C -- 9.2. Conservative operations -- 9.3. Simple conditions -- 9.4. Measurability of convex functions -- 9.5. Plane curves -- 9.6. Skew curves -- 9.7. Boundedness below -- 9.8. Restrictions of convex functions and additive functions -- 10. The Classes U, B, C -- 10.1. A Hahn-Banach theorem -- 10.2. The class B -- 10.3. The class C -- 10.4. The class U -- 10.5. Set-theoretic operations -- 10.6. The classes D -- 10.7. The classes U[subscript C] and B[subscript C] -- 11. Properties of Hamel Bases -- 11.1. General properties -- 11.2. Measure -- 11.3. Topological properties -- 11.4. Burstin bases -- 11.5. Erdos sets -- 11.6. Lusin sets -- 11.7. Perfect sets -- 11.8. The operations R and U -- 12. Further Properties of Additive Functions and Convex Functions -- 12.1. Graphs -- 12.2. Additive functions -- 12.3. Convex functions -- 12.4. Big graph -- 12.5. Invertible additive functions -- 12.6. Level sets -- 12.7. Partitions -- 12.8. Monotonicity -- Part III. Related Topics -- 13. Related Equations -- 13.1. The remaining Cauchy equations -- 13.2. Jensen equation -- 13.3. Pexider equations -- 13.4. Multiadditive functions -- 13.5. Cauchy equation on an interval -- 13.6. The restricted Cauchy equation -- 13.7. Hosszu equation -- 13.8. Mikusinski equation -- 13.9. An alternative equation -- 13.10. The general linear equation -- 14. Derivations and Automorphisms -- 14.1. Derivations -- 14.2. Extensions of derivations -- 14.3. Relations between additive functions -- 14.4. Automorphisms of R -- 14.5. Automorphisms of C -- 14.6. Non-trivial endomorphisms of C -- 15. Convex Functions of Higher Orders -- 15.1. The difference operator -- 15.2. Divided differences -- 15.3. Convex functions of higher order -- 15.4. Local boundedness of p-convex functions -- 15.5. Operation H -- 15.6. Continuous p-convex functions -- 15.7. Continuous p-convex functions. Case N = 1 -- 15.8. Differentiability of p-convex functions -- 15.9. Polynomial functions -- 16. Subadditive Functions -- 16.1. General properties -- 16.2. Boundedness. Continuity -- 16.3. Differentiability -- 16.4. Sublinear functions -- 16.5. Norm -- 16.6. Infinitary subadditive functions -- 17. Nearly Additive Functions and Nearly Convex Functions -- 17.1. Approximately additive functions -- 17.2. Approximately multiadditive functions -- 17.3. Functions with bounded differences -- 17.4. Approximately convex functions -- 17.5. Set ideals -- 17.6. Almost additive functions -- 17.7. Almost polynomial functions -- 17.8. Almost convex functions -- 17.9. Almost subadditive functions -- 18. Extensions of Homomorphisms -- 18.1. Commutative divisible groups -- 18.2. The simplest case of S generating X -- 18.3. A generalization -- 18.4. Further extension theorems -- 18.5. Cauchy equation on a cylinder -- 18.6. Cauchy nucleus -- 18.7. Theorem of Ger -- 18.8. Inverse additive functions -- 18.9. Concluding remarks -- Bibliography -- Indices -- Index of Symbols -- Subject Index -- Index of Names. Publisher Marketing: Marek Kuczma was born in 1935 in Katowice, Poland, and died there in 1991. After finishing high school in his home town, he studied at the Jagiellonian University in Krakow. He defended his doctoral dissertation under the supervision of Stanislaw Golab. In the year of his habilitation, in 1963, he obtained a position at the Katowice branch of the Jagiellonian University (now University of Silesia, Katowice), and worked there till his death. Besides his several administrative positions and his outstanding teaching activity, he accomplished excellent and rich scientific work publishing three monographs and 180 scientific papers. He is considered to be the founder of the celebrated Polish school of functional equations and inequalities. "The second half of the title of this book describes its contents adequately. Probably even the most devoted specialist would not have thought that about 300 pages can be written just about the Cauchy equation (and on some closely related equations and inequalities). And the book is by no means chatty, and does not even claim completeness. Part I lists the required preliminary knowledge in set and measure theory, topology and algebra. Part II gives details on solutions of the Cauchy equation and of the Jensen inequality [...], in particular on continuous convex functions, Hamel bases, on inequalities following from the Jensen inequality [...]. Part III deals with related equations and inequalities (in particular, Pexider, Hosszu, and conditional equations, derivations, convex functions of higher order, subadditive functions and stability theorems). It concludes with an excursion into the field of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews) "This book is a real holiday for all the mathematicians independently of their strict speciality. One can imagine what deliciousness represents this book for functional equationists." (B. Crstici, Zentralblatt fur Mathematik) "