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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 Department of Mathematics University of Michigan Ann Arbor, Michigan, USA Lawrence Zalcman Department of Mathematics Bar-Ilan University 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... Publications of M. M. Schiffer... 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. Charzyński) 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 v ix 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 für Orthogonalinvarianten. Math. Z. 38 (1934), 315 322. 2. Sur un principe nouveau pour l évaluation 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 problème d extrémum de la représentation 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 théorie des transformations pseudoconformes. C. R. Acad. Sci. Paris 207 (1938), 112 115. 8. Sur un théorèmedelareprésentation conforme. C. R. Acad. Sci. Paris 207 (1938), 520 522. 9. (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), 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 diamètre 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. 65 (1943), 341 360. 14. The span of multiply connected domains. Duke Math. J. 10 (1943), 209 216. 15. (with S. Bergman) Bounded functions of two complex variables. Amer.J.Math.66 (1944), 161 169. 16. Sur l équation différentielle de M. Löwner. C. R. Acad. Sci. Paris 221 (1945), 369 371. 17. Hadamard s formula and variation of domain-functions. Amer.J.Math.68 (1946), 417 448. 18. On the modulus of doubly-connected domains. Quart. J. Math. Oxford Ser. 17 (1946), 197 213. 19. The kernel function of an orthonormal system. Duke Math. J. 13 (1946), 529 540. 20. (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), 609 638. 21. (with S. Bergman) On Green s and Neumann s functions in the theory of partial differential equations. Bull. Amer. Math. Soc. 53 (1947), 1141 1151. 22. An application of orthonormal functions in the theory of conformal mapping. Amer. J. Math. 70 (1948), 147 156. 23. (with S. Bergman) Kernel functions in the theory of partial differential equations of elliptic type. Duke Math. J. 15 (1948), 535 566. ix

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. 249 285. 25. Faber polynomials in the theory of univalent functions. Bull. Amer. Math. Soc. 54 (1948), 503 517. 26. (with P. R. Garabedian) Identities in the theory of conformal mapping. Trans. Amer. Math. Soc. 65 (1949), 187 238. 27. (with G. Szegő) Virtual mass and polarization. Trans. Amer. Math. Soc. 67 (1949), 130 205. 28. (with A. C. Schaeffer and D. C. Spencer) The coefficient regions of schlicht functions. Duke Math. J. 16 (1949), 493 527. 29. (with P. R. Garabedian) On existence theorems of potential theory and conformal mapping. Ann. of Math. 52 (1950), 164 187. 30. (with D. C. Spencer) The coefficient problem for multiply-connected domains. Ann. of Math. (2) 52 (1950), 362 402. 31. Various types of orthogonalization. Duke Math. J. 17 (1950), 329 366. 32. (with S. Bergman) Various kernels in the theory of partial differential equations. Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 559 563. 33. (with S. Bergman) Some linear operators in the theory of partial differential equations. Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 742 746. 34. 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. 249 323. 35. (with S. Bergman) Kernel functions and conformal mapping. Compositio Math. 8 (1951), 205 249. 36. (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), 375 386. 37. (with S. Bergman) A majorant method for non-linear partial differential equations. Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 744 749. 38. (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. 233 240. 41. (with K. S. Miller) On the Green s function of ordinary differential systems. Proc. Amer. Math. Soc. 3 (1952), 433 441. 42. (with K. S. Miller) Monotonic properties of the Green s function. Proc. Amer. Math. Soc. 3 (1952), 948 956. 43. (with F. S. Bodenheimer) Mathematical studies in animal populations. I. A mathematical study of insect parasitism. Acta Biotheoretica Ser. A 10 (1952), 23 56. 44. (with P. R. Garabedian and H. Lewy) Axially symmetric cavitational flow. Ann. of Math. (2) 56 (1952), 560 602. 45. (with S. Bergman) Potential-theoretic methods in the theory of functions of two complex variables. Compositio Math. 10 (1952), 213 240. 46. (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. 193 198. 47. (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. 199 206. 48. (with P. R. Garabedian) Variational problems in the theory of elliptic partial differential equations. J. Rational Mech. Anal. 2 (1953), 137 171. 49. (with P. R. Garabedian) Convexity of domain functionals. J. Analyse Math. 2 (1953), 281 368.

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. 15 30. 51. (with S. Bergman) Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, New York, 1953. 52. The Applied Mathematics Laboratory at Stanford University, in Proceedings of a Conference on Training in Applied Mathematics (1953), pp. 37 40. 53. (with G. Pólya; appendix by H. Helfenstein) Convexity of functionals by transplantation. J. Analyse Math. 3 (1954), 245 346. 54. Variation of domain functionals. Bull. Amer. Math. Soc. 60 (1954), 303 328. 55. (with D. C. Spencer) Functionals of Finite Riemann Surfaces, Princeton University Press, Princeton, N.J., 1954. 56. (with P. R. Garabedian) On estimation of electrostatic capacity. Proc. Amer. Math. Soc. 5 (1954), 206 211. 57. (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. 79 87. 58. (with P. R. Garabedian) On a double integral variational problem. Canad. J. Math. 6 (1954), 441 446. 59. (with P. R. Garabedian) A coefficient inequality for schlicht functions. Ann. of Math. (2) 61 (1955), 116 136. 60. (with P. R. Garabedian) A proof of the Bieberbach conjecture for the fourth coefficient. J. Rational Mech. Anal. 4 (1955), 427 465. 61. Partial differential equations of elliptic type, in Modern Mathematics for the Engineer (E. F. Beckenbach, editor), McGraw-Hill, New York, 1956, pp. 110 144. 62. The Fredholm eigen values of plane domains. Pacific J. Math. 7 (1957), 1187 1225. 63. Sur les rapports entre les solutions des problèmes intérieurs et celles des problèmes extérieurs. C. R. Acad. Sci. Paris 244 (1957), 2680 2683. 64. Sur la polarisation et la masse virtuelle. C. R. Acad. Sci. Paris 244 (1957), 3118 3121. 65. Problèmes aux limites et fonctions propres de l équation intégrale de Poincaré et de Fredholm. C. R. Acad. Sci. Paris 245 (1957), 18 21. 66. 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. 97 149. 67. 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. 93 113. 68. Fredholm eigen values of multiply-connected domains. Pacific J. Math. 9 (1959), 211 269. 69. (with G. Pólya) Sur la représentation conforme de l extérieur d une courbe fermée convexe. C. R. Acad. Sci. Paris 248 (1959), 2837 2839. 70. 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. 211 231. 71. Analytical theory of subsonic and supersonic flows, in Handbuch der Physik, Springer-Verlag, Berlin, 1960, Vol. 9, Part 3, pp. 1 161. 72. (with Z. Charzyński) A new proof of the Bieberbach conjecture for the fourth coefficient. Arch. Rational Mech. Anal. 5 (1960), 187 193. 73. (with Z. Charzyński) A geometric proof of the Bieberbach conjecture for the fourth coefficient. Scripta Math. 25 (1960), 173 181. 74. (with N. S. Hawley) Connections and conformal mapping. Acta Math. 107 (1962), 175 274. 75. (with P. L. Duren) A variational method for functions schlicht in an annulus. Arch. Rational Mech. Anal. 9 (1962), 260 272. 76. (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. 341 358. 77. (with P. L. Duren) The theory of the second variation in extremum problems for univalent functions. J. Analyse Math. 10 (1962/63), 193 252.

xii Publications of M. M. Schiffer 78. Fredholm eigenvalues and conformal mapping. Rend. Mat. (5) 22 (1963), 447 468. 79. (with E. Reich) Estimates for the transfinite diameter of a continuum. Math. Z. 85 (1964), 91 106. 80. (with R. Adler and M. Bazin) Introduction to General Relativity, McGraw-Hill, New York, 1965. 81. (with B. Epstein) On the mean-value property of harmonic functions. J. Analyse Math. 14 (1965), 109 111. 82. (with G. Springer) Fredholm eigenvalues and conformal mapping of multiply connected domains. J. Analyse Math. 14 (1965), 337 378. 83. (with O. Tammi) On the fourth coefficient of bounded univalent functions. Trans. Amer. Math. Soc. 119 (1965), 67 78. 84. (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), 115 134. 86. (with P. R. Garabedian and G. G. Ross) On the Bieberbach conjecture for even n. J. Math. Mech. 14 (1965), 975 989. 87. (with N. S. Hawley) Half-order differentials on Riemann surfaces. Acta Math. 115 (1966), 199 236. 88. Half-order differentials on Riemann surfaces. SIAM J. Appl. Math. 14 (1966), 922 934. 89. (with O. Tammi) A method of variations for functions with bounded boundary rotation. J. Analyse Math. 17 (1966), 109 144. 90. A variational method for univalent quasiconformal mappings. Duke Math. J. 33 (1966), 395 411. 91. (with P. R. Garabedian) The local maximum theorem for the coefficients of univalent functions. Arch. Rational Mech. Anal. 26 (1967), 1 32. 92. (with N. S. Hawley) Riemann surfaces which are doubles of plane domains. Pacific J. Math. 20 (1967), 217 222. 93. (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), 329 349. 95. On the coefficient problem for univalent functions. Trans. Amer. Math. Soc. 134 (1968), 95 101. 96. (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), 461 474. 98. (with J. A. Hummel) Coefficient inequalities for Bieberbach-Eilenberg functions. Arch. Rational Mech. Anal. 32 (1969), 87 99. 99. Some distortion theorems in the theory of conformal mapping. Atti Accad. Naz. Lincei Mem.Cl.Sci.Fis.Mat.Natur.Sez.Ia(8) 10 (1970), 1 19. 100. (with R. N. Pederson) Further generalizations of the Grunsky inequalities. J. Analyse Math. 23 (1970), 353 380. 101. (with H. G. Schmidt) A new set of coefficient inequalities for univalent functions. Arch. Rational Mech. Anal. 42 (1971), 346 368. 102. (with O. Tammi) A Green s inequality for the power matrix. Ann. Acad. Sci. Fenn. Ser. A I 501 (1971), 15 pp. 103. (with G. Schober) An extremal problem for the Fredholm eigenvalues. Arch. Rational Mech. Anal. 44 (1971/72), 83 92. 104. (with R. N. Pederson) A proof of the Bieberbach conjecture for the fifth coefficient. Arch. Rational Mech. Anal. 45 (1972), 161 193. 105. Inequalities in the theory of univalent functions, in Inequalities III (O. Shisha, editor), Academic Press, New York, 1972, pp. 311 319. 106. (with G. Schober) A remark on the paper An extremal problem for the Fredholm eigenvalues. Arch. Rational Mech. Anal. 46 (1972), 394. 107. (with R. J. Adler, J. Mark, and C. Sheffield) Kerr geometry as complexified Schwarzschild geometry. J. Mathematical Phys. 14 (1973), 52 56. 108. Article in How to Write Mathematics, American Mathematical Society, Providence, R.I., 1973, pp. 49 61.

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), 99 114. 110. (with R. Osserman) Doubly-connected minimal surfaces. Arch. Rational Mech. Anal. 58 (1975), 285 307. 111. (with G. Schober) Coefficient problems and generalized Grunsky inequalities for schlicht functions with quasiconformal extensions. Arch. Rational Mech. Anal. 60 (1975/76), 205 228. 112. (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. 138 147. 113. (with S. Friedland) Global results in control theory with applications to univalent functions. Bull. Amer. Math. Soc. 82 (1976), 913 915. 114. (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), 501 531. 115. (with S. Friedland) On coefficient regions of univalent functions. J. Analyse Math. 31 (1977), 125 168. 116. (with J. A. Hummel) Variational methods for Bieberbach-Eilenberg functions and for pairs. Ann. Acad. Sci. Fenn. Ser. A I Math. 3 (1977), 3 42. 117. (with G. Schober) A variational method for general families of quasiconformal mappings. J. Analyse Math. 34 (1978), 240 264. 118. (with H. Samelson) Dedicated to the memory of Stefan Bergman. Applicable Anal. 8 (1978/79), 195 199 (1 plate). 119. (with E. Netanyahu) On the monotonicity of some functionals in the family of univalent functions. Israel J. Math. 32 (1979), 14 26. 120. (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. 349 357. 121. (with J. A. Hummel and B. Pinchuk) Bounded univalent functions which cover a fixed disc. J. Analyse Math. 36 (1979), 118 138. 122. (with G. Schober) The dielectric Green s function and quasiconformal mapping. J. Analyse Math. 36 (1979), 233 243. 123. (with A. Chang and G. Schober) On the second variation for univalent functions. J. Analyse Math. 40 (1981), 203 238. 124. Stefan Bergman (1895 1977): in memoriam. Ann. Polon. Math. 39 (1981), 5 9 (1 plate). 125. Fredholm eigenvalues and Grunsky matrices. Ann. Polon. Math. 39 (1981), 149 164. 126. (with D. Aharonov and L. Zalcman) Potato kugel. Israel J. Math. 40 (1981), 331 339. 127. (with P. L. Duren and Y. J. Leung) Support points with maximum radial angle. Complex Variables Theory Appl. 1 (1982/83), 263 277. 128. (with L. Bowden) The Role of Mathematics in Science, Mathematical Association of America (New Mathematical Library, Vol. 30), Washington, D. C., 1984. 129. George Pólya (1887 1985). Math. Mag. 60 (1987), 268 270. 130. Pólya s contributions in mathematical physics, Bull. London Math. Soc. 19 (1987), 591 594. [Part of obituary of George Pólya, ibid., pp. 559 608.] 131. (with P. L. Duren) Grunsky inequalities for univalent functions with prescribed Hayman index. Pacific J. Math. 131 (1988), 105 117. 132. (with P. L. Duren) Conformal mappings onto nonoverlapping regions, in Complex Analysis (J. Hersch and A. Huber, editors), Birkhäuser Verlag, Basel, 1988, pp. 27 39. 133. (with P. L. Duren) Sharpened forms of the Grunsky inequalities. J. Analyse Math. 55 (1990), 96 116. 134. (with P. L. Duren) Goluzin inequalities and minimum energy for mappings onto nonoverlapping regions. Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), 133 150. 135. (with P. L. Duren) Univalent functions which map onto regions of given transfinite diameter. Trans. Amer. Math. Soc. 323 (1991), 413 428. 136. (with P. L. Duren) Robin functions and energy functionals of multiply connected domains. Pacific J. Math. 148 (1991), 251 273.

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), 189 196. 138. Issai Schur: Some personal reminiscences, in Mathematik in Berlin, Vol. II, (H. Begehr, editor), Shaker Verlag, Aachen, 1998, pp. 177 181. 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, 118 119. (with S. Sambursky) Static universe and nebular red shift. II, Phys. Rev. (2) 53 (1938), 256 263.