The Theory of Inequalities began its development from the time when C. F. GACSS, A. L. CATCHY and P. L. CEBYSEY, to mention only the most important, laid the theoretical foundation for approximative meth­ ods. Around the end of the 19th and the beginning of the 20th century, numerous inequalities were proyed, some of which became classic, while most remained as isolated and unconnected results. It is almost generally acknowledged that the classic work "Inequali­ ties" by G. H. HARDY, J. E. LITTLEWOOD and G. POLYA, which appeared in 1934, transformed the field of inequalities from a collection of isolated formulas into a systematic discipline. The modern Theory of Inequalities, as well as the continuing and growing interest in this field, undoubtedly stem from this work. The second English edition of this book, published in 1952, was unchanged except for three appendices, totalling 10 pages, added at the end of the book. Today inequalities playa significant role in all fields of mathematics, and they present a very active and attractive field of research. J. DIEUDONNE, in his book "Calcullnfinitesimal" (Paris 1968), attri­ buted special significance to inequalities, adopting the method of exposi­ tion characterized by "majorer, minorer, approcher". Since 1934 a multitude of papers devoted to inequalities have been published: in some of them new inequalities were discovered, in others classical inequalities ,vere sharpened or extended, various inequalities ,vere linked by finding their common source, while some other papers gave a large number of miscellaneous applications.

Inhalt

1. Introduction.- 1.1 Real Number System.- 1.1.1 Axioms of the Set of Real Numbers.- 1.1.2 Order Properties of Real Numbers.- 1.2 Complex Number System.- 1.3 Monotone Functions.- 1.4 Convex Functions.- 1.4.1 Definitions of Jensen Convex Functions.- 1.4.2 Continuity of Jensen Convex Functions.- 1.4.3 Convex Functions.- 1.4.4 Continuity and Differentiability of Convex Functions.- 1.4.5 Logarithmically Convex Functions.- 1.4.6 Some Extensions of the Concept of Convex Functions.- 1.4.7 Hierarchy of Convexity.- 2. General Inequalities.- 2.1 Fundamental Inequalities.- 2.1.1 Simple Means.- 2.1.2 Cauchy's Inequality.- 2.2 Abel's Inequality.- 2.3 Jordan's Inequality.- 2.4 Bernoulli's Inequality and its Generalizations.- 2.5 ?ebySev's and Related Inequalities.- 2.6 Cauchy's and Related Inequalities.- 2.6.1 Some Refinements and Extensions of Cauchy's Inequality.- 2.6.2 Gram's Inequality.- 2.7 Young's Inequality.- 2.8 Hölder's Inequality.- 2.9 Minkowski's and Related Inequalities.- 2.10 Inequalities of Aczél, Popoviciu, Kurepa and Bellman.- 2.11 Schweitzer's, Diaz-Metcalf's, Rennie's and Related Inequalities.- 2.12 An Inequality of Fan and Todd.- 2.13 Grüss' Inequality.- 2.14 Means.- 2.14.1 Definitions.- 2.14.2 Inequalities Involving Means.- 2.14.3 Ratios and Differences of Means.- 2.14.4 Refinement of the Arithmetic-Geometric Mean Inequality.- 2.14.5 Some General Inequalities Involving Means.- 2.14.6 The ?-Method of Mitrinovi? and Vasi?.- 2.15 Symmetric Means and Functions.- 2.15.1 Definitions and Main Relations between Symmetric Means.- 2.15.2 Inequalities of Rado-Popoviciu Type.- 2.15.3 Concavity of Certain Functions Involving the Elementary Symmetric Functions.- 2.16 Steffensen's and Related Inequalities.- 2.17 Schur's Inequality.- 2.18 Turán's Inequalities.- 2.19 Benson's Method.- 2.20 Recurrent Inequalities of Redheffer.- 2.21 Cyclic Inequalities.- 2.22 Inequalities Involving Derivatives.- 2.23 Integral Inequalities Involving Derivatives.- 2.23.1 An Inequality Ascribed to Wirtinger.- 2.23.2 An Inequality of Opial.- 2.24 Inequalities Connected with Majorization of Vectors.- 2.25 Inequalities for Vector Norms.- 2.25.1 Triangle Inequality.- 2.25.2 An Identity of Hlawka and the Associated Inequality.- 2.25.3 An Inequality of Hornich.- 2.25.4 Generalizations of Hlawka's Inequality.- 2.25.5 A Steinitz-Gross Result.- 2.26 Mills Ratio and Some Related Results.- 2.27 Stirling's Formula.- 3. Particular Inequalities.- 3.1 Inequalities Involving Functions of Discrete Variables.- 3.2 Inequalities Involving Algebraic Functions.- 3.3 Inequalities Involving Polynomials.- 3.4 Inequalities Involving Trigonometric Functions.- 3.5 Inequalities Involving Trigonometric Polynomials.- 3.6 Inequalities Involving Exponential, Logarithmic and Gamma Functions.- 3.7 Integral Inequalities.- 3.8 Inequalities in the Complex Domain.- 3.9 Miscellaneous Inequalities.- Name Index.