Inequalities Based on Sobolev Representations
Inequalities based on Sobolev Representations deals exclusively with very general tight integral inequalities of Chebyshev-Grüss, Ostrowski types and of integral means, all of which depend upon the Sobolev integral representations of functions. Appl
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George A. Anastassiou
Inequalities Based on Sobolev Representations
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George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152 USA [email protected]
ISSN 2191-8198 e-ISSN 2191-8201 ISBN 978-1-4614-0200-8 e-ISBN 978-1-4614-0201-5 DOI 10.1007/978-1-4614-0201-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011929870 Mathematics Subject Classification (2010): 26D10, 26D15, 26D20 c George A. Anastassiou All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To the memory of my friend Khalid Khouri who left this world too young The measure of success for a person is the magnitude of his/her ability to convert negative conditions to positive ones and achieve goals —The author
Preface
This brief monograph is the first one to deal exclusively with very general tight integral inequalities of Chebyshev–Gr¨uss, Ostrowski types, and of the comparison of integral means. These rely on the well-known Sobolev integral representations of functions. The inequalities engage ordinary and weak partial derivatives of the involved functions. Applications of these developments are illustrated. On the way to prove the main results we derive important estimates for the averaged Taylor polynomials and remainders of Sobolev integral representations. The exposed results expand to all possible directions. We examine both the univariate and multivariate cases. For the convenience of the reader, each chapter of this book is written in a selfcontained style. This treatise relies on the author’s last year of related research work. Advanced courses and seminars can be taught out of this brief book. All necessary background and motivations are given in each chapter. A related list of references is also given at the end of each chapter. These results first appeared in my articles that are mentioned in the references. The results are expected to find applications in many subareas of mathematical analysis, inequalities, partial differential equations, information theory, etc. As such this monograph is suitable for researchers, graduate students, seminars of the above subjects, and also for all science libraries. The p
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