Since the classic work on inequalities by Hardy, Littlewood, and Polya in 1934, an enormous amount of effort has been devoted to the sharpening and extension of the classical inequalities, to the discovery of new types of inequalities, and to the application of inqualities in many parts of analysis. As examples, let us cite the fields of ordinary and partial differenlial equations, which aie dominated by inequalities and variational principles involving functions and their derivatives; the many applications of linear inequalities to game theory and mathematical economics, which have triggered a renewed interest in convexity and moment-space theory; and the growing uses of digital computers, which have given impetus to a systematic study of error estimates involving much sophisticated matrix theory and operator theory.