Contemporary Mathematicians

Transcription

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2 Contemporary Mathematicians Gian-Carlo Rota Joseph P.S. Kung Editors For further volumes:

3 Peter Duren Lawrence Zalcman Editors Menahem Max Schiffer: Selected Papers Volume 2

4 Editors Peter Duren Department of Mathematics University of Michigan Ann Arbor, Michigan, USA Lawrence Zalcman Department of Mathematics Bar-Ilan University Ramat-Gan, Israel ISBN ISBN (ebook) DOI / Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 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 (

5 Max Schiffer in his Stanford office, ca.1976

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7 Contents Frontispiece... Publications of M. M. Schiffer... The Fredholm eigen values of plane domains... 1 Fredholm eigen values of multiply-connected domains Fredholm eigenvalues and conformal mapping Fredholm eigenvalues and Grunsky matrices Commentary by Reiner Kühnau (with G. Pólya) Sur la représentation conforme de l extérieur d une courbe fermée convexe Commentary by Peter Duren Extremum problems and variational methods in conformal mapping Commentary by Peter Duren (with Z. Charzyński) A new proof of the Bieberbach conjecture for the fourth coefficient Commentary by Peter Duren (with P. L. Duren) A variational method for functions schlicht in an annulus Commentary by Peter Duren (with B. Epstein) On the mean-value property of harmonic functions Commentary by Lawrence Zalcman (with N. S. Hawley) Half-order differentials on Riemann surfaces Commentary by John Fay (with P. R. Garabedian) The local maximum theorem for the coefficients of univalent functions Commentary by Peter Duren Some distortion theorems in the theory of conformal mapping Commentary by Peter Duren v ix vii

8 viii Contents (with G. Schober) An extremal problem for the Fredholm eigenvalues (with G. Schober) A remark on the paper An extremal problem for the Fredholm eigenvalues (with G. Schober) A variational method for general families of quasiconformal mappings Commentary by Reiner Kühnau (with J. Hersch and L. E. Payne) Some inequalities for Stekloff eigenvalues Commentary by Bodo Dittmar (with J. A. Hummel) Variational methods for Bieberbach Eilenberg functions and for pairs Commentary by Dov Aharonov (with J. A. Hummel and B. Pinchuk) Bounded univalent functions which cover a fixed disc Commentary by Bernard Pinchuk (with G. Schober) The dielectric Green s function and quasiconformal mapping Commentary by Brad Osgood (with A. Chang and G. Schober) On the second variation for univalent functions Commentary by Peter Duren (with D. Aharonov and L. Zalcman) Potato kugel Commentary by Lawrence Zalcman (with P. L. Duren and Y. J. Leung) Support points with maximum radial angle Commentary by Peter Duren (with P. L. Duren) Univalent functions which map onto regions of given transfinite diameter Commentary by Peter Duren (with P. L. Duren) Robin functions and distortion of capacity under conformal mapping Commentary by Peter Duren Issai Schur: Some personal reminiscences Commentary by Lawrence Zalcman

9 Publications of M. M. Schiffer 1. Ein neuer Beweis des Endlichkeitssatzes für Orthogonalinvarianten. Math. Z. 38 (1934), Sur un principe nouveau pour l évaluation des fonctions holomorphes. Bull. Soc. Math. France 64 (1936), Un calcul de variation pour une famille de fonctions univalentes. C. R. Acad. Sci. Paris 205 (1937), Sur un problème d extrémum de la représentation conforme. Bull. Soc. Math. France 66 (1938), A method of variation within the family of simple functions. Proc. London Math. Soc. (2) 44 (1938), On the coefficients of simple functions. Proc. London Math. Soc. (2) 44 (1938), Sur les domaines minima dans la théorie des transformations pseudoconformes. C. R. Acad. Sci. Paris 207 (1938), Sur un théorèmedelareprésentation conforme. C. R. Acad. Sci. Paris 207 (1938), (with S. Bergmann) Familles bornées de fonctions de deux variables complexes dans des domaines avec une surface remarquable. C. R. Acad. Sci. Paris 207 (1938), Sur la variation de la fonction de Green de domaines plans quelconques. C. R. Acad. Sci. Paris 209 (1939), Sur la variation du diamètre transfini. Bull. Soc. Math. France 68 (1940), On the subadditivity of the transfinite diameter. Proc. Cambridge Philos. Soc. 37 (1941), Variation of the Green function and theory of the p-valued functions. Amer.J.Math. 65 (1943), The span of multiply connected domains. Duke Math. J. 10 (1943), (with S. Bergman) Bounded functions of two complex variables. Amer.J.Math.66 (1944), Sur l équation différentielle de M. Löwner. C. R. Acad. Sci. Paris 221 (1945), Hadamard s formula and variation of domain-functions. Amer.J.Math.68 (1946), On the modulus of doubly-connected domains. Quart. J. Math. Oxford Ser. 17 (1946), The kernel function of an orthonormal system. Duke Math. J. 13 (1946), (with S. Bergman) A representation of Green s and Neumann s functions in the theory of partial differential equations of second order. Duke Math. J. 14 (1947), (with S. Bergman) On Green s and Neumann s functions in the theory of partial differential equations. Bull. Amer. Math. Soc. 53 (1947), An application of orthonormal functions in the theory of conformal mapping. Amer. J. Math. 70 (1948), (with S. Bergman) Kernel functions in the theory of partial differential equations of elliptic type. Duke Math. J. 15 (1948), ix

10 x Publications of M. M. Schiffer 24. (with R. von Mises) On Bergman s integration method in two-dimensional compressible fluid flow, in Advances in Applied Mechanics (R. von Mises and T. von Kármán, editors), Academic Press, New York, 1948, pp Faber polynomials in the theory of univalent functions. Bull. Amer. Math. Soc. 54 (1948), (with P. R. Garabedian) Identities in the theory of conformal mapping. Trans. Amer. Math. Soc. 65 (1949), (with G. Szegő) Virtual mass and polarization. Trans. Amer. Math. Soc. 67 (1949), (with A. C. Schaeffer and D. C. Spencer) The coefficient regions of schlicht functions. Duke Math. J. 16 (1949), (with P. R. Garabedian) On existence theorems of potential theory and conformal mapping. Ann. of Math. 52 (1950), (with D. C. Spencer) The coefficient problem for multiply-connected domains. Ann. of Math. (2) 52 (1950), Various types of orthogonalization. Duke Math. J. 17 (1950), (with S. Bergman) Various kernels in the theory of partial differential equations. Proc. Nat. Acad. Sci. U.S.A. 36 (1950), (with S. Bergman) Some linear operators in the theory of partial differential equations. Proc. Nat. Acad. Sci. U.S.A. 36 (1950), Some recent developments in the theory of conformal mapping, in R. Courant, Dirichlet s Principle, Conformal Mapping, and Minimal Surfaces, Interscience, New York, 1950, Appendix, pp (with S. Bergman) Kernel functions and conformal mapping. Compositio Math. 8 (1951), (with S. Bergman) Kernel functions and partial differential equations. I. Boundary value problems in the theory of non-linear partial differential equations of elliptic type. J. Analyse Math. 1 (1951), (with S. Bergman) A majorant method for non-linear partial differential equations. Proc. Nat. Acad. Sci. U.S.A. 37 (1951), (with D. C. Spencer) A variational calculus for Riemann surfaces. Ann. Acad. Sci. Fenn. Ser. A I 93 (1951), 9 pp. 39. (with D. C. Spencer) On the conformal mapping of one Riemann surface into another. Ann. Acad. Sci. Fenn. Ser. A I 94 (1951), 10 pp. 40. Variational methods in the theory of conformal mapping, in Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, American Mathematical Society, Providence, R.I., 1952, Vol. 2, pp (with K. S. Miller) On the Green s function of ordinary differential systems. Proc. Amer. Math. Soc. 3 (1952), (with K. S. Miller) Monotonic properties of the Green s function. Proc. Amer. Math. Soc. 3 (1952), (with F. S. Bodenheimer) Mathematical studies in animal populations. I. A mathematical study of insect parasitism. Acta Biotheoretica Ser. A 10 (1952), (with P. R. Garabedian and H. Lewy) Axially symmetric cavitational flow. Ann. of Math. (2) 56 (1952), (with S. Bergman) Potential-theoretic methods in the theory of functions of two complex variables. Compositio Math. 10 (1952), (with D. C. Spencer) Some remarks on variational methods applicable to multiply connected domains, in Construction and Applications of Conformal Maps: Proceedings of a Symposium (E. F. Beckenbach, editor), National Bureau of Standards, Appl. Math. Series 18, U.S. Government Printing Office, Washington, D.C., 1952, pp (with S. Bergman) Theory of kernel functions in conformal mapping, in Construction and Applications of Conformal Maps: Proceedings of a Symposium (E. F. Beckenbach, editor), National Bureau of Standards, Appl. Math. Series 18, U.S. Government Printing Office, Washington, D.C., 1952, pp (with P. R. Garabedian) Variational problems in the theory of elliptic partial differential equations. J. Rational Mech. Anal. 2 (1953), (with P. R. Garabedian) Convexity of domain functionals. J. Analyse Math. 2 (1953),

11 Publications of M. M. Schiffer xi 50. Variational methods in the theory of Riemann surfaces, in Contributions to the Theory of Riemann Surfaces (L. Ahlfors et al., editors), Annals of Math. Studies 30, Princeton University Press, Princeton, N.J., 1953, pp (with S. Bergman) Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, New York, The Applied Mathematics Laboratory at Stanford University, in Proceedings of a Conference on Training in Applied Mathematics (1953), pp (with G. Pólya; appendix by H. Helfenstein) Convexity of functionals by transplantation. J. Analyse Math. 3 (1954), Variation of domain functionals. Bull. Amer. Math. Soc. 60 (1954), (with D. C. Spencer) Functionals of Finite Riemann Surfaces, Princeton University Press, Princeton, N.J., (with P. R. Garabedian) On estimation of electrostatic capacity. Proc. Amer. Math. Soc. 5 (1954), (with S. Bergman) Properties of solutions of a system of partial differential equations, in Studies in Mathematics and Mechanics Presented to Richard von Mises, Academic Press, New York, 1954, pp (with P. R. Garabedian) On a double integral variational problem. Canad. J. Math. 6 (1954), (with P. R. Garabedian) A coefficient inequality for schlicht functions. Ann. of Math. (2) 61 (1955), (with P. R. Garabedian) A proof of the Bieberbach conjecture for the fourth coefficient. J. Rational Mech. Anal. 4 (1955), Partial differential equations of elliptic type, in Modern Mathematics for the Engineer (E. F. Beckenbach, editor), McGraw-Hill, New York, 1956, pp The Fredholm eigen values of plane domains. Pacific J. Math. 7 (1957), Sur les rapports entre les solutions des problèmes intérieurs et celles des problèmes extérieurs. C. R. Acad. Sci. Paris 244 (1957), Sur la polarisation et la masse virtuelle. C. R. Acad. Sci. Paris 244 (1957), Problèmes aux limites et fonctions propres de l équation intégrale de Poincaré et de Fredholm. C. R. Acad. Sci. Paris 245 (1957), Partial differential equations of the elliptic type, in Lecture Series of the Symposium on Partial Differential Equations, University of California, Berkeley, 1955 (N. Aronszajn and C. B. Morrey, Jr., editors), University of Kansas Press, Lawrence, Kansas, 1957, pp Applications of variational methods in the theory of conformal mapping, in Calculus of Variations and its Applications, Proc. Symposia Appl. Math. 8 (L. M. Graves, editor), McGraw-Hill, New York, 1958, pp Fredholm eigen values of multiply-connected domains. Pacific J. Math. 9 (1959), (with G. Pólya) Sur la représentation conforme de l extérieur d une courbe fermée convexe. C. R. Acad. Sci. Paris 248 (1959), Extremum problems and variational methods in conformal mapping, in Proceedings of the International Congress of Mathematicians, Edinburgh, 1958, Cambridge University Press, New York, 1960, pp Analytical theory of subsonic and supersonic flows, in Handbuch der Physik, Springer-Verlag, Berlin, 1960, Vol. 9, Part 3, pp (with Z. Charzyński) A new proof of the Bieberbach conjecture for the fourth coefficient. Arch. Rational Mech. Anal. 5 (1960), (with Z. Charzyński) A geometric proof of the Bieberbach conjecture for the fourth coefficient. Scripta Math. 25 (1960), (with N. S. Hawley) Connections and conformal mapping. Acta Math. 107 (1962), (with P. L. Duren) A variational method for functions schlicht in an annulus. Arch. Rational Mech. Anal. 9 (1962), (with J. Siciak) Transfinite diameter and analytic continuation of functions of two complex variables, in Studies in Mathematical Analysis and Related Topics; Essays in Honor of George Pólya (G. Szegő, editor), Stanford University Press, Stanford, California, 1962, pp (with P. L. Duren) The theory of the second variation in extremum problems for univalent functions. J. Analyse Math. 10 (1962/63),

12 xii Publications of M. M. Schiffer 78. Fredholm eigenvalues and conformal mapping. Rend. Mat. (5) 22 (1963), (with E. Reich) Estimates for the transfinite diameter of a continuum. Math. Z. 85 (1964), (with R. Adler and M. Bazin) Introduction to General Relativity, McGraw-Hill, New York, (with B. Epstein) On the mean-value property of harmonic functions. J. Analyse Math. 14 (1965), (with G. Springer) Fredholm eigenvalues and conformal mapping of multiply connected domains. J. Analyse Math. 14 (1965), (with O. Tammi) On the fourth coefficient of bounded univalent functions. Trans. Amer. Math. Soc. 119 (1965), (with O. Tammi) The fourth coefficient of a bounded real univalent function. Ann. Acad. Sci. Fenn. Ser. A I 354 (1965), 32 pp. 85. (with L. Sario and M. Glasner) The span and principal functions in Riemannian spaces. J. Analyse Math. 15 (1965), (with P. R. Garabedian and G. G. Ross) On the Bieberbach conjecture for even n. J. Math. Mech. 14 (1965), (with N. S. Hawley) Half-order differentials on Riemann surfaces. Acta Math. 115 (1966), Half-order differentials on Riemann surfaces. SIAM J. Appl. Math. 14 (1966), (with O. Tammi) A method of variations for functions with bounded boundary rotation. J. Analyse Math. 17 (1966), A variational method for univalent quasiconformal mappings. Duke Math. J. 33 (1966), (with P. R. Garabedian) The local maximum theorem for the coefficients of univalent functions. Arch. Rational Mech. Anal. 26 (1967), (with N. S. Hawley) Riemann surfaces which are doubles of plane domains. Pacific J. Math. 20 (1967), (with O. Tammi) On the fourth coefficient of univalent functions with bounded boundary rotation. Ann. Acad. Sci. Fenn. Ser. A I 396 (1967), 26 pp. 94. Univalent functions whose n first coefficients are real. J. Analyse Math. 18 (1967), On the coefficient problem for univalent functions. Trans. Amer. Math. Soc. 134 (1968), (with O. Tammi) On bounded univalent functions which are close to identity. Ann. Acad. Sci. Fenn. Ser. A I 435 (1968), 26 pp. 97. (with O. Tammi) On the coefficient problem for bounded univalent functions. Trans. Amer. Math. Soc. 140 (1969), (with J. A. Hummel) Coefficient inequalities for Bieberbach-Eilenberg functions. Arch. Rational Mech. Anal. 32 (1969), Some distortion theorems in the theory of conformal mapping. Atti Accad. Naz. Lincei Mem.Cl.Sci.Fis.Mat.Natur.Sez.Ia(8) 10 (1970), (with R. N. Pederson) Further generalizations of the Grunsky inequalities. J. Analyse Math. 23 (1970), (with H. G. Schmidt) A new set of coefficient inequalities for univalent functions. Arch. Rational Mech. Anal. 42 (1971), (with O. Tammi) A Green s inequality for the power matrix. Ann. Acad. Sci. Fenn. Ser. A I 501 (1971), 15 pp (with G. Schober) An extremal problem for the Fredholm eigenvalues. Arch. Rational Mech. Anal. 44 (1971/72), (with R. N. Pederson) A proof of the Bieberbach conjecture for the fifth coefficient. Arch. Rational Mech. Anal. 45 (1972), Inequalities in the theory of univalent functions, in Inequalities III (O. Shisha, editor), Academic Press, New York, 1972, pp (with G. Schober) A remark on the paper An extremal problem for the Fredholm eigenvalues. Arch. Rational Mech. Anal. 46 (1972), (with R. J. Adler, J. Mark, and C. Sheffield) Kerr geometry as complexified Schwarzschild geometry. J. Mathematical Phys. 14 (1973), Article in How to Write Mathematics, American Mathematical Society, Providence, R.I., 1973, pp

13 Publications of M. M. Schiffer xiii 109. (with J. Hersch and L. E. Payne) Some inequalities for Stekloff eigenvalues. Arch. Rational Mech. Anal. 57 (1975), (with R. Osserman) Doubly-connected minimal surfaces. Arch. Rational Mech. Anal. 58 (1975), (with G. Schober) Coefficient problems and generalized Grunsky inequalities for schlicht functions with quasiconformal extensions. Arch. Rational Mech. Anal. 60 (1975/76), (with G. Schober) A distortion theorem for quasiconformal mappings, in Advances in Complex Function Theory, Maryland 1973/74 (W.E.KirwanandL.Zalcman, editors), Lecture Notes in Math. 505, Springer-Verlag, Berlin, 1976, pp (with S. Friedland) Global results in control theory with applications to univalent functions. Bull. Amer. Math. Soc. 82 (1976), (with G. Schober) Representation of fundamental solutions for generalized Cauchy- Riemann equations by quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), (with S. Friedland) On coefficient regions of univalent functions. J. Analyse Math. 31 (1977), (with J. A. Hummel) Variational methods for Bieberbach-Eilenberg functions and for pairs. Ann. Acad. Sci. Fenn. Ser. A I Math. 3 (1977), (with G. Schober) A variational method for general families of quasiconformal mappings. J. Analyse Math. 34 (1978), (with H. Samelson) Dedicated to the memory of Stefan Bergman. Applicable Anal. 8 (1978/79), (1 plate) (with E. Netanyahu) On the monotonicity of some functionals in the family of univalent functions. Israel J. Math. 32 (1979), (with G. Schober) An application of the calculus of variations for general families of quasiconformal mappings, in Complex Analysis, Joensuu 1978 (I. Laine, O. Lehto, and T. Sorvali, editors), Lecture Notes in Math. 747, Springer-Verlag, Berlin, 1979, pp (with J. A. Hummel and B. Pinchuk) Bounded univalent functions which cover a fixed disc. J. Analyse Math. 36 (1979), (with G. Schober) The dielectric Green s function and quasiconformal mapping. J. Analyse Math. 36 (1979), (with A. Chang and G. Schober) On the second variation for univalent functions. J. Analyse Math. 40 (1981), Stefan Bergman ( ): in memoriam. Ann. Polon. Math. 39 (1981), 5 9 (1 plate) Fredholm eigenvalues and Grunsky matrices. Ann. Polon. Math. 39 (1981), (with D. Aharonov and L. Zalcman) Potato kugel. Israel J. Math. 40 (1981), (with P. L. Duren and Y. J. Leung) Support points with maximum radial angle. Complex Variables Theory Appl. 1 (1982/83), (with L. Bowden) The Role of Mathematics in Science, Mathematical Association of America (New Mathematical Library, Vol. 30), Washington, D. C., George Pólya ( ). Math. Mag. 60 (1987), Pólya s contributions in mathematical physics, Bull. London Math. Soc. 19 (1987), [Part of obituary of George Pólya, ibid., pp ] 131. (with P. L. Duren) Grunsky inequalities for univalent functions with prescribed Hayman index. Pacific J. Math. 131 (1988), (with P. L. Duren) Conformal mappings onto nonoverlapping regions, in Complex Analysis (J. Hersch and A. Huber, editors), Birkhäuser Verlag, Basel, 1988, pp (with P. L. Duren) Sharpened forms of the Grunsky inequalities. J. Analyse Math. 55 (1990), (with P. L. Duren) Goluzin inequalities and minimum energy for mappings onto nonoverlapping regions. Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), (with P. L. Duren) Univalent functions which map onto regions of given transfinite diameter. Trans. Amer. Math. Soc. 323 (1991), (with P. L. Duren) Robin functions and energy functionals of multiply connected domains. Pacific J. Math. 148 (1991),

14 xiv Publications of M. M. Schiffer 137. (with P. L. Duren) Robin functions and distortion of capacity under conformal mapping. Complex Variables Theory Appl. 21 (1993), Issai Schur: Some personal reminiscences, in Mathematik in Berlin, Vol. II, (H. Begehr, editor), Shaker Verlag, Aachen, 1998, pp Editors Note. Two items, which appear in lists of Schiffer s publications prepared in the 1940s but are absent from later lists, have been omitted: Lösung der Aufgabe 128, Jber. Deutsch. Math.-Verein. 42 (1933), Abt. 2, (with S. Sambursky) Static universe and nebular red shift. II, Phys. Rev. (2) 53 (1938),