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This monograph presents univariate and multivariate classical analyses of advanced inequalities. This treatise is a culmination of the author's last thirteen years of research work. The chapters are self-contained and several advanced courses can be taught out of this book. Extensive background and motivations are given in each chapter with a comprehensive list of references given at the end. The topics covered are wide-ranging and diverse. Recent advances on Ostrowski type inequalities, Opial type inequalities, Poincare and Sobolev type inequalities, and Hardy-Opial type inequalities are exam

Inequalities (Mathematics)

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Discrete Hilbert-type inequalities including Hilbert's inequality are important in mathematical analysis and its applications. In 1998, the author presented an extension of Hilbert's integral inequality with an independent parameter. In 2004, some new extensions of Hilbert's inequality were presented by introducing two pairs of conjugate exponents and additional independent parameters. Since then, a number of new discrete Hilbert-type inequalities have arisen. In this book, the author explains how to use the way of weight coefficients and introduce specific parameters to build new discrete Hil

Inequalities (Mathematics) --- Processes, Infinite

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Inequalities (Mathematics) --- Processes, Infinite

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Most people, when they think of mathematics, think first of numbers and equations-the number (x)=that number (y). But professional mathematicians, in dealing with quantities that can be ordered according to their size, often are more interested in unequal magnitudes that are equal. This book provides an introduction to the fascinating world of inequalities beginning with a systematic discussion of the relation 'greater than' and the meaning of 'absolute values' of numbers, and ending with descriptions of some unusual geometries. In the course of the book, the reader will encounter some of the more famous inequalities in mathematics. Starting with the basic order properties of real numbers, this book carries the reader through the classical inequalities of Cauchy, Minkowsky and Hörder with many variants and applications. The concluding chapter points the way to other metrics in the plane and the interrelations between geometry (convexity) and algebra (inequalities).

Inequalities (Mathematics) --- Processes, Infinite.

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Collecting all the results on the particular types of inequalities, the coverage of this book is unique among textbooks in the literature. The book focuses on the historical development of the Carlson inequalities and their many generalizations and variations. As well as almost all known results concerning these inequalities and all known proof techniques, a number of open questions suitable for further research are considered. Two chapters are devoted to clarifying the close connection between interpolation theory and this type of inequality. Other applications are also included, in addition

Inequalities (Mathematics) --- Interpolation. --- Numerical analysis.