Sharp Real-Part Theorems A Unified Approach
This volume contains a coherent point of view on various sharp pointwise inequalities for analytic functions in a disk in terms of the real part of the function on the boundary circle or in the disk itself. Inequalities of this type are frequently used in
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Gershon Kresin Vladimir G. Maz’ya
Sharp Real-Part Theorems A Unified Approach
1903
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1903
Gershon Kresin · Vladimir Maz’ya
Sharp Real-Part Theorems A Unified Approach Translated from Russian and edited by T. Shaposhnikova
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Authors Vladimir Maz’ya Department of Mathematics Ohio State University 231 W 18th Avenue Columbus, OH 43210 USA e-mail: [email protected]
Gershon Kresin Department of Computer Science and Mathematics College of Judea and Samaria 44837 Ariel Israel e-mail: [email protected]
Department of Mathematical Sciences University of Liverpool M&O Building Liverpool L69 3BX UK e-mail: [email protected] and Department of Mathematics Link¨oping University 58183 Link¨oping Sweden e-mail: [email protected] Library of Congress Control Number: 2006940398 Mathematics Subject Classification (2000): 30A10, 30B10, 31A05 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-69573-7 Springer Berlin Heidelberg New York ISBN-13 978-3-540-69573-8 Springer Berlin Heidelberg New York DOI 10.1007/3-540-69573-7 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, 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. Typesetting by the authors and SPi using a Springer LATEX macro package Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper
SPIN: 11970422
VA41/3100/SPi
543210
Preface
We present a unified approach to various sharp pointwise inequalities for analytic functions in a disk with the real part of the function on the circumference as the right-hand side. We refer to these inequalities as “real-part theorems” in reference to the first assertion of such a kind, the celebrated Hadamard’s real-part theorem (1892). The inequalities in question are frequently used in the theory of entire functions and in the analytic number theory. We hope that collecting these inequalities in one place, as well as generalizing and refining them, may prove useful for various applications. In particular, one can anticipate rich opportunities to extend these inequalities to analytic functions of several complex variables and solutions of partial
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