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Proof of a conjecture of F. Nevanlinna concerning functions which have deficiency sum two. (English) Zbl 0622.30028

While developing ”Nevanlinna theory” the Nevanlinna brothers formulated many conjectures. All of these were settled by now except for F. Nevanlinna’s conjecture that a meromorphic function f(z) of finite order \(\lambda\) and with \(\sum_{a\in {\hat {\mathbb{C}}}}\delta (a,f)=2\) must satisfy \[ (1)\quad^ 2\lambda \text{ is an integer \(\geq 2\) and (2) }\delta (a,f)=p(a)/\lambda,\quad p(a)\in {\mathbb{Z}}, \] and all deficient values are asymptotic.
The paper under review shows that this conjecture is correct. The principal tool is the theory of quasiconformal modifications of meromorphic functions developed by the author and A. Weitsman [see D. Drasin, Acta Math. 138, 83-151 (1977; Zbl 0355.30028)]. The proof requires many delicate, intricately linked arguments leading up to the construction of a quasiconformal modification of \(f(z^ 2)\) which behaves essentially like \(\exp (z^{2\lambda})\) and from whose properties the truth of the conjecture follows.
Reviewer: W.H.J.Fuchs

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D30 Meromorphic functions of one complex variable (general theory)

Citations:

Zbl 0355.30028
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References:

[1] Ahlfors, L. V., Zur Theorie der Überlagerungsflachen.Acta Math., 64 (1935), 157–194. · Zbl 0012.17204
[2] –,Lectures on quasiconformal mappings. Van Nostrand, Princeton, 1967.
[3] Boas, R. P.,Entire functions. Academic Press, New York, 1954. · Zbl 0058.30201
[4] Drasin, D., Quasi-conformal modifications of functions having deficiency sum two.Ann. of Math., 114 (1981), 493–518. · Zbl 0469.30022
[5] –, A note on functions with deficiency sum two.Ann. Acad. Sci. Fenn. (A.I.), 2 (1976), 59–66. · Zbl 0346.30020
[6] –, The inverse problem of the Nevanlinna theory.Acta Math., 138 (1977), 83–151. · Zbl 0355.30028
[7] Drasin, D. &Shea, D. F., Pólya peaks and the oscillation of positive functions.Proc. Amer. Math. Soc., 34 (1972), 403–411. · Zbl 0258.26004
[8] Drasin, D &Weitsman, A., Meromorphic functions with large sums of deficiencies.Adv. in Math., 15 (1975), 93–126. · Zbl 0294.30022
[9] Duren, P. L.,Theory of H p Spaces, Academic Press, New York, 1970. · Zbl 0215.20203
[10] Edrei, A. &Fuchs, W. H. J., On the growth of meromorphic functions with several deficient values.Trans. Amer. Math. Soc., 93 (1959), 292–328. · Zbl 0092.07201
[11] –, Valeurs déficientes et valeurs asymptotiques des fonctions meromorphes.Comment. Math. Helv., 33 (1959), 258–295. · Zbl 0090.28802
[12] –, Bounds for the number of deficient values of certain classes of meromorphic functions.Proc. London Math. Soc. (3), 12 (1962), 315–344. · Zbl 0103.30001
[13] Fuchs, W. H. J., A theorem on the Nevanlinna deficiencies of meromorphic functions of finite order.Ann. of Math., 68 (1958), 203–209. · Zbl 0083.06601
[14] –, Proof of a conjecture of G. Pólya concerning gap series.Illinois J. Math., 7 (1963), 661–667. · Zbl 0113.28702
[15] –,Topics in the theory of funtions of one complex variable. Van Nostrand, Princeton, 1967.
[16] –,Topics in Nevanlinna theory. Proc. NRL Conf. on Classical function theory. Naval Research Laboratory, Washington, 1970. · Zbl 0294.30021
[17] Goldberg, A. A. &Ostrovskii, I. V.,The Distribution of values of meromorphic functions (in Russian). Nauka, Moscow, 1970.
[18] Hayman, W. K.,Meromorphic functions. Oxford, 1964. · Zbl 0115.06203
[19] Hille, E.,Lectures on ordinary differential equations. Addison Wesley, Reading (Mass.), 1969. · Zbl 0179.40301
[20] Miles, J., A note on Ahlfors’ theory covering surfaces.Proc. Amer. Math. Soc., 21 (1969), 30–32. · Zbl 0172.37002
[21] Nevanlinna, F., Über eine Klass meromorpher Funktionen.Comptes rendus de septième congrès des mathématiciens scandinaves tenu à Oslo 19–22 août 1929. A. W. Brøggers boktrykkeri. A/S, Oslo, 1930.
[22] Nevanlinna, R.,Le théorème de Picard-Borel et la théorie des fonctions méromorphes. Gauthier Villars, Paris, 1929. · JFM 55.0773.03
[23] –,Analytic functions. Springer, New York, 1970. · Zbl 0199.19801
[24] Pfluger, A., Zur Defektrelation ganzer Funktionen endicher Ordnung.Comment Math. Helv., 19 (1946), 91–104. · Zbl 0063.06209
[25] Rickman, S., Solution to the problem of Picard’s theorem for quasiregular mappings in dimension three. To appear. · Zbl 0617.30024
[26] Weitsman, A., Meromorphic functions with maximal deficiency sum and a conjecture of F. Nevanlinna.Acta Math., 123 (1969), 115–139. · Zbl 0185.14502
[27] –, A growth property of the Nevanlinna characteristic.Proc. Amer. Math. Soc., 26 (1970), 65–70. · Zbl 0201.09302
[28] –, A theorem on Nevanlinna deficiencies.Acta Math., 128 (1972), 41–52. · Zbl 0229.30028
[29] –,Some remarks on the spread of a Nevanlinna deficiency. Report No. 7 (1977), Institut Mittag-Leffler, Djursholm (Sweden).
[30] Wittich, H.,Neuere Untersuchungen über eindeutige analytische Funktionen. Springer, Berlin, 1955. · Zbl 0067.05501
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