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
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lIerausgegeben von
J. L. Doob . A. Grothendieck
. E. Heinz . F. Hirzebruch E.Hopf. H. Hopf. W. Maak . S. Mac Lane· W. Magnus M. M. Postnikov . F. K. Schmidt· D. S. Scott· K. Stein
C;eschaJft~uhrende
lIerausgeber
B. Eckmann und B. L. van der Waerden
D. S. Mitrinovic
Analytic Inequalities
In cooperation with
P.M.Vasic
Springer-Verlag Berlin· Heidelberg. New York 1970
Dr. Dragosla\' S. Mitrinovic Professor of Mathematics at the Belgrade t:niversity Belgrade, Yugoslavia
Dr. Petar M. Vasic Assistant Professor at the Belgrade University Belgrade, Yugoslavia
Ge3chaftsiiihrende Hera usgeber:
Professor Dr. B. Eckmann Eidgenossische Technische Hochschule Zurich
Professor Dr. B. L. yan der \Vaerden Mathematisches Institut der Universitat Zurich
ISBN-13: 978-3-642-99972-7 e-ISBN-13: 978-3-642-99970-3 DOl: 10.1007/978-3-642-99970-3
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. by Springer'\'erlag, Berlin· Heidelberg 19iO. Softcover reprint of the hardcover 1st edition 1970 Library of Congress Catalog Card Kumber 76-116492 Title No. 5148
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Preface 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 methods. 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 "Inequalities" 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), attributed special significance to inequalities, adopting the method of exposition 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 n
