Contemporary Mathematicians Peter Duren Lawrence Zalcman Editors Menahem Max Schiffer Selected Papers Volume 2 Contemporary Mathematicians Gian-Carlo Rota† Joseph P.S. Kung Editors For further volumes: http://www.springer.com/series/4817 Peter Duren • Lawrence Zalcman Editors Menahem Max Schiffer: Selected Papers Volume 2 Editors Peter Duren Lawrence Zalcman Department of Mathematics Department of Mathematics University of Michigan Bar-Ilan University Ann Arbor, Michigan, USA Ramat-Gan, Israel ISBN 978-1-4614-7948-2 ISBN 978-1-4614-7949-9 (eBook) DOI 10.1007/978-1-4614-7949-9 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2013948450 Mathematics Subject Classification (2010): 30-XX, 31-XX, 35-XX, 49-XX, 76-XX, 20-XX, 01-XX © Springer Science+Business Media New York 2014 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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com) Max Schiffer in his Stanford office, ca.1976 Contents Frontispiece .............................................................................................. v Publications of M. M. Schiffer ................................................................ ix The Fredholm eigen values of plane domains .......................................... 1 Fredholm eigen values of multiply-connected domains ........................... 41 Fredholm eigenvalues and conformal mapping ........................................ 101 Fredholm eigenvalues and Grunsky matrices ........................................... 125 Commentary by Reiner Kühnau ........................................................... 142 (with G. Pólya) Sur la représentation conforme de l’extérieur d’une courbe fermée convexe ................................................................... 145 Commentary by Peter Duren ................................................................ 149 Extremum problems and variational methods in conformal mapping ..... 151 Commentary by Peter Duren ................................................................ 173 (with Z. Charzyn´ski) A new proof of the Bieberbach conjecture for the fourth coefficient ........................................................................... 175 Commentary by Peter Duren ................................................................ 183 (with P. L. Duren) A variational method for functions schlicht in an annulus .............................................................................................. 185 Commentary by Peter Duren ................................................................ 199 (with B. Epstein) On the mean-value property of harmonic functions ..... 201 Commentary by Lawrence Zalcman .................................................... 205 (with N. S. Hawley) Half-order differentials on Riemann surfaces ......... 207 Commentary by John Fay ..................................................................... 246 (with P. R. Garabedian) The local maximum theorem for the coefficients of univalent functions ............................................................ 247 Commentary by Peter Duren ................................................................ 280 Some distortion theorems in the theory of conformal mapping ............... 283 Commentary by Peter Duren ................................................................ 302 vii viii Contents (with G. Schober) An extremal problem for the Fredholm eigenvalues ................................................................................................ 303 (with G. Schober) A remark on the paper “An extremal problem for the Fredholm eigenvalues” .................................................................. 315 (with G. Schober) A variational method for general families of quasiconformal mappings ..................................................................... 317 Commentary by Reiner Kühnau ........................................................... 343 (with J. Hersch and L. E. Payne) Some inequalities for Stekloff eigenvalues ............................................................................................... 345 Commentary by Bodo Dittmar ............................................................. 362 (with J. A. Hummel) Variational methods for Bieberbach—Eilenberg functions and for pairs .............................................................................. 365 Commentary by Dov Aharonov ........................................................... 406 (with J. A. Hummel and B. Pinchuk) Bounded univalent functions which cover a fixed disc ........................................................................... 409 Commentary by Bernard Pinchuk ........................................................ 431 (with G. Schober) The dielectric Green’s function and quasiconformal mapping ........................................................................... 433 Commentary by Brad Osgood .............................................................. 445 (with A. Chang and G. Schober) On the second variation for univalent functions .................................................................................... 447 Commentary by Peter Duren ................................................................ 484 (with D. Aharonov and L. Zalcman) Potato kugel .................................... 487 Commentary by Lawrence Zalcman .................................................... 497 (with P. L. Duren and Y. J. Leung) Support points with maximum radial angle ................................................................................................ 499 Commentary by Peter Duren ................................................................ 515 (with P. L. Duren) Univalent functions which map onto regions of given transfinite diameter ..................................................................... 517 Commentary by Peter Duren ................................................................ 534 (with P. L. Duren) Robin functions and distortion of capacity under conformal mapping ......................................................................... 537 Commentary by Peter Duren ................................................................ 546 Issai Schur: Some personal reminiscences ................................................ 549 Commentary by Lawrence Zalcman .................................................... 555 Publications of M. M. Schiffer 1. Ein neuer Beweis des Endlichkeitssatzes f¨ur Orthogonalinvarianten. Math. Z. 38 (1934), 315–322. 2. Sur un principe nouveau pour l’evaluation´ des fonctions holomorphes. Bull. Soc. Math. France 64 (1936), 231–240. 3. Un calcul de variation pour une famille de fonctions univalentes. C. R. Acad. Sci. Paris 205 (1937), 709–711. 4. Sur un probleme` d’extremum´ de la representation´ conforme. Bull. Soc. Math. France 66 (1938), 48–55. 5. A method of variation within the family of simple functions. Proc. London Math. Soc. (2) 44 (1938), 432–449. 6. On the coefficients of simple functions. Proc. London Math. Soc. (2) 44 (1938), 450– 452. 7. Sur les domaines minima dans la theorie´ des transformations pseudoconformes. C. R. Acad. Sci. Paris 207 (1938), 112–115. 8. Sur un theor´ emedelarepr` esentation´ conforme. C. R. Acad. Sci. Paris 207 (1938), 520–522. 9. (with S. Bergmann) Familles bornees´ de fonctions de deux variables complexes dans des domaines avec une surface remarquable. C. R. Acad. Sci. Paris 207 (1938), 711– 713. 10. Sur la variation de la fonction de Green de domaines plans quelconques. C. R. Acad. Sci. Paris 209 (1939), 980–982. 11. Sur la variation du diametre` transfini. Bull. Soc. Math. France 68 (1940), 158–176. 12. On the subadditivity of the transfinite diameter. Proc. Cambridge Philos. Soc. 37 (1941), 373–383. 13. Variation of the Green function and theory of the p-valued functions. Amer.J.Math.