Analytic Inequalities

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.- 1.2 Complex Number System.- 1.3 Monotone Functions.- 1.4 Convex Functions.- 2. General Inequalities.- 2.1 Fundamental Inequalities.- 2.2 Abel's Inequality.- 2.3 Jordan's Inequality.- 2.4 Bernoulli's Inequality and its Generalizations.- 2.5 ?ebyev's and Related Inequalities.- 2.6 Cauchy's and Related Inequalities.- 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.- 2.15 Symmetric Means and 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.- 2.24 Inequalities Connected with Majorization of Vectors.- 2.25 Inequalities for Vector Norms.- 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.