Hardy Type Inequalities on Time Scales
The book is devoted to dynamic inequalities of Hardy type and extensions and generalizations via convexity on a time scale T. In particular, the book contains the time scale versions of classical Hardy type inequalities, Hardy and Littlewoo
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dy Type Inequalities on Time Scales
Hardy Type Inequalities on Time Scales
Ravi P. Agarwal • Donal O’Regan • Samir H. Saker
Hardy Type Inequalities on Time Scales
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Ravi P. Agarwal Department of Mathematics Texas A&M University–Kingsville Kingsville, TX, USA
Donal O’Regan School of Mathematics Statistics and Applied Mathematics National University of Ireland Galway, Ireland
Samir H. Saker Department of Mathematics Mansoura University Mansoura, Egypt
ISBN 978-3-319-44298-3 ISBN 978-3-319-44299-0 (eBook) DOI 10.1007/978-3-319-44299-0 Library of Congress Control Number: 2016950725 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Ravi P. Agarwal: To Sadhna, Sheba, and Danah Donal O’Regan: To Alice, Aoife, Lorna, Daniel, and Niamh Samir H. Saker: To Mona, Meran, Maryam, Mennah, and Ahmed
Preface
Neither of us completely understood what the other was doing, but we realized that our joint effort will give the theorem, and to be a little impudent and conceited, probabilistic number theory was born! This collaboration is a good example to show that two brains can be better than one, since neither of us could have done the work alone. Paul Erd˝os (1933–1996)
Hardy’s interest in inequalities (in both discrete and continuous forms) was during the period 1906–1928. As a result of his work, the subject was changed radically, and what had previously been a collection of isolated formulas became a systematic discipline. The classical book Inequalities by Hardy et al. [77] contains two chapters devoted to Hardy- and Hilbert-type inequalities and the growth of Hardy-type inequalities in the literature stimulated this book. The book is devoted to dynamic inequalities of Hardy type and extensions and generaliz
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