Abstract
Zürich is a special place to workers in meromorphic function theory. Rolf Nevanlinna was Professor both at the ETH and University of Zürich. His address at the 1932 Zürich ICM centered on connections between his new theory of meromorphic functions and the Riemann surface off-1, a perspective that continues to yield insights. Lars Ahlfors accompanied Nevanlinna to the ETH in 1928, where he developed his fundamental distortion theorem and proved Denjoy’s conjecture that an entire function of order ρ has at most 2ρ distinct finite asymptotic values. Zürich has been one of the main venues of the Nevanlinna Colloquia through the years, and the home of Pólya and Pfluger.
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References
N. U. Arakelyan,Entire functions of finite order with an infinite set of deficient values, Dokl. Akad. Nauk. SSSR170 (1966), 999–1002 (in Russian); Engl, transl.: Soviet Math Dokl.7 (1966), 1303–1306.
N. U. Arakelyan,Entire functions of finite order with an infinite set of deficient values, Dokl. Akad. Nauk. SSSR170 (1966), 999–1002 (in Russian); Engl, transl.: Soviet Math Dokl.7 (1966), 1303–1306.
A. Baernstein,Proof of Edrei’s spread conjecture, Proc. London Math. Soc26 (1973), 418–434.
I. N. Baker,Repulsive fixpoints of entire functions, Math. Z.104 (1968), 252–256.
W. Bergweiler and A. Eremenko,On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana, to appear..
D. Drasin,The inverse problem of the Nevanlinna theory, Acta Math.138 (1977), 83–151.
D. Drasin,Proof of a conjecture of F. Nevanlinna concerning functions which have deficiency sum two, Acta Math.158 (1987), 1–94.
D. Drasin and A. Weitsman,Meromorphic functions with large sums of deficiencies, Adv. in Math.15 (1975), 93–126.
E. A. Edrei,Locally Tauberian theorems for meromorphic functions of lower order less than one, Trans. Amer. Math. Soc.140 (1969), 309–332.
E. A. Edrei,Solution of the deficiency problem for functions of small lower order, Proc. London Math. Soc.26 (1973), 435–445.
E. A. Edrei and W. H. J. Fuchs,Valeurs déficientes et valuers asymptotiques des fonctions méromorphes, Comm. Math. Helv.33 (1959), 258–295.
A. E. Eremenko,The inverse problem of the theory of meromorphic functions of finite order, Sibirsk. Mat. Zh.27 (1986), 377–390.
A. E. Eremenko,A new proof of Drasin’s theorem on meromorphic functions of finite order with maximal deficiency sum, I and II, Teor. Funktsi Funktionals Anal, i Prilozhen. (Kharkov) 51 (1989), 107–116; 52 (1990), 3522–3527;52 (1990), 3397–3403.
A. E. Eremenko,A new proof of Drasin’s theorem on meromorphic functions of finite order with maximal deficiency sum, I and II, Teor. Funktsiǐ Funktionals Anal, i Prilozhen. (Kharkov)51 (1989), 107–116;52 (1990), 3522–3527;52 (1990), 3397–3403.
A. E. Eremenko,A new proof of Drasin’s theorem on meromorphic functions of finite order with maximal deficiency sum, I and II, Teor. Funktsiǐ Funktionals Anal, i Prilozhen. (Kharkov)51 (1989), 107–116;52 (1990), 3522–3527;52 (1990), 3397–3403.
A. E. Eremenko,A counterexample to the Arakelyan conjecture, Bull. Amer. Math. Soc.27 (1992), 159–164.
A. E. Eremenko,Meromorphic functions with small ramification, Indiana Univ. Math. J.42 (1993), 1193–1218.
A. E. Eremenko and J. L. Lewis,Uniform limits of certain A-harmonic functions with applications to quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math.16 (1991), 361–375.
A. E. Eremenko and M. Sodin,On meromorphic functions of finite order with maximal deficiency sum, Teor. Funktsiǐ, Funktionals Anal, i Prilozhen. (Kharkov)59 (1992), 643–651.
A. E. Eremenko and M. Sodin,Distribution of values of meromorphic functions and meromorphic curves from the point of view of potential theory, Algebra i Analiz3 (1991), 131–164 (inRussian); Engl, trans.: St. Petersburg Math. J.3 (1991), 109–136.
A. E. Eremenko,Distribution of values of meromorphic functions and meromorphic curves from the point of view of potential theory, Algebra i Analiz3 (1991), 131–164 (inRussian); Engl, trans.: St. Petersburg Math. J.3 (1991), 109–136.
W. H. J. Fuchs,A theorem on the Nevanlinna deficiencies of meromorphic functions of finite order, Ann. of Math. (2)68 (1958), 203–209.
W. H. J. and W. K. Hayman,An entire function with assigned deficiencies, Stud. Math. Analysis and Related Topics, Stanford Univ. Press, Stanford, CA, (1962), 117–125.
A. A. Goldberg,Meromorphic functions, in v. 10 Serija Math. Analiz, Itogi Nauki i Tekhniki, Moscow (1973) (in Russian); Engl, trans.: J. Soviet Math.4 (1975), 157–216.
A. A. Goldberg and V. A. Grinshtein,The logarithmic derivative of a meromorphic function, Mat. Zametki19 (1976), 525–530 (in Russian); Engl, transl.: Math. Notes19 (1976), 320–323.
A. A. Goldberg and V. A. Grinshtein,The logarithmic derivative of a meromorphic function, Mat. Zametki19 (1976), 525–530 (in Russian); Engl, transl.: Math. Notes19 (1976), 320–323.
A. A. Goldberg, B. Ja. Levin, and I. V. Ostrovskii,Entire and meromorphic functions, Kompleksaii Analyiz Odna Peremennaja-1, Tom 85;Serija Sovremennie problemi mathematiki, Itogi Nauki i Tekhniki, Moscow, 1991.
W. K. Hayman, Meromorphic Functions, Oxford University Press, 1964.
W. K. Hayman and J. Miles,On the growth of a meromorphic function and its derivatives, Complex Variables12 (1989), 245–260.
A. Hinkkanen,A sharp form of Nevanlinna’s second fundamental theorem, Invent. Math.108 (1992), 549–574.
I. Holopainen and S. Rickman,A Picard type theorem for quasiregular mappings of ℝn into n-manifolds with many ends, Rev. Mat. Iberoamericana8 (1992), 131–148.
S. Lang,The error term in Nevanlinna theory, Duke Math J.56 (1988), 193–218.
J. Langley,On the deficiencies of composite entire functions, Proc. Edinburgh Math. Soc.36 (1992), 151–164.
J. L. Lewis and J.-M. Wu,On conjectures of Arakelyan and Littlewood, J. Analyse Math.50 (1988), 259–283.
F. Nevanlinna,Über eine Klasse meromorpher Funktionen, C.R. 7e Congr. Math. Scand. Oslo (1929), 81–83.
R. Nevanlinna, Le théorème de Picard-Borel et la théorie des fonctions méromorphes, Gauthier-Villars, Paris, 1929.
Ch. Osgood,Sometimes effective Thue-Siegel-Roth-Nevanlinna bounds, or better, J. Number Theory21 (1985), 347–399.
X. Pang,On normal criterion of meromorphic functions, Sci. Sinica (5)33 (1990), 521–527.
A. Pfluger,Zur Defekt relation ganzer Funktionen endlicher Ordnung, Comm. Math. Helv.19 (1946), 91–104.
S. Rickman, Quasiregular Mappings, Ergeb. Math. Grenzgeb.,26 (1993), Springer- Verlag, Berlin and New York.
M. Ru and P-M. Wong,Integral points of ℙn \ 2n + 1hyperplanes in general position, Invent. Math.106 (1991), 195–216.
J. Schiff, Universitext, Springer-Verlag, Berlin and New York, 1993.
W. Schwick,Repelling points in the Julia set,, to appear, Bull. London Math. Soc.
D. Shea,On the frequency of multiple values of a meromorphic function of small order, Michigan Math. J.32 (1985), 109–116.
Y. Sibuya, Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient, North-Holland, Amsterdam, 1975.
N. Steinmetz,Eine Verallgemeinerung des zweiten Nevanlinnaschen Hauptsatzes, J. Reine Angew. Math.368 (1986), 131–141.
P. Vojta, Diophantine approximations and value distribution theory, Springer-Verlag, Berlin and New York, 1987.
A. Weitsman,Asymptotic behavior of meromorphic functions with extremal deficiencies, Trans. Amer. Math. Soc.140 (1969), 333–352.
______,A theorem on Nevanlinna deficiencies, Acta. Math.128 (1972), 41–52.
P.-M. Wong,On the second main theorem in Nevanlinna theory, Amer. J. Math.111 (1989), 549–583.
L. Zalcman,Normal families revisited, Complex Analysis and Related Topics (J.J.O.O. Weigerinck, ed. ), Univ. of Amsterdam, 1993.
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© 1995 Birkhäuser Verlag, Basel, Switzerland
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Drasin, D. (1995). Meromorphic Functions: Progress and Problems. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_76
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