An Introduction to the Theory of Functional Equations and Inequalities, Second Edition, edited by Attila Gilányi, is a comprehensive monograph on functional equations and inequalities. The book is based on a course given by Marek Kuczma at the Silesian University in 1974/75. It covers Cauchy's equation and Jensen's inequality, which are central to the theory of functional equations. The second edition, published in 2009, retains the structure and content of the first edition, with some minor changes marked by footnotes. The book is divided into three parts: Preliminaries, Cauchy's Functional Equation and Jensen's Inequality, and Related Topics. The first part includes chapters on set theory, topology, measure theory, and algebra, providing the necessary background for understanding functional equations and inequalities. The second part focuses on Cauchy's equation and Jensen's inequality, discussing additive and convex functions, their properties, and related inequalities. The third part covers related equations, derivations and automorphisms, convex functions of higher orders, and subadditive functions. The book is intended for mathematicians and students interested in functional equations and inequalities. It provides a systematic exposition of the theory, with exercises and bibliographical hints to aid further study. The text emphasizes the theory rather than examples or applications, though some applications of the Cauchy equation are mentioned. The book assumes a basic knowledge of calculus, Lebesgue measure and integral, algebra, topology, and set theory. It includes a detailed index of symbols, subjects, and names, making it a valuable reference for researchers and students in mathematics.An Introduction to the Theory of Functional Equations and Inequalities, Second Edition, edited by Attila Gilányi, is a comprehensive monograph on functional equations and inequalities. The book is based on a course given by Marek Kuczma at the Silesian University in 1974/75. It covers Cauchy's equation and Jensen's inequality, which are central to the theory of functional equations. The second edition, published in 2009, retains the structure and content of the first edition, with some minor changes marked by footnotes. The book is divided into three parts: Preliminaries, Cauchy's Functional Equation and Jensen's Inequality, and Related Topics. The first part includes chapters on set theory, topology, measure theory, and algebra, providing the necessary background for understanding functional equations and inequalities. The second part focuses on Cauchy's equation and Jensen's inequality, discussing additive and convex functions, their properties, and related inequalities. The third part covers related equations, derivations and automorphisms, convex functions of higher orders, and subadditive functions. The book is intended for mathematicians and students interested in functional equations and inequalities. It provides a systematic exposition of the theory, with exercises and bibliographical hints to aid further study. The text emphasizes the theory rather than examples or applications, though some applications of the Cauchy equation are mentioned. The book assumes a basic knowledge of calculus, Lebesgue measure and integral, algebra, topology, and set theory. It includes a detailed index of symbols, subjects, and names, making it a valuable reference for researchers and students in mathematics.