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Normal families: New perspectives


Author: Lawrence Zalcman
Journal: Bull. Amer. Math. Soc. 35 (1998), 215-230
MSC (1991): Primary 30D45; Secondary 30D35, 34A20, 58F23
MathSciNet review: 1624862
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Abstract: This paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains. These include short and efficient proofs of generalizations of (i) the Picard Theorems, (ii) Gol'dberg's Theorem (a meromorphic function on $\mathbb{C}$ which is the solution of a first-order algebraic differential equation has finite order), and (iii) the Fatou-Julia Theorem (the Julia set of a rational function of degree $d\ge 2$ is the closure of the repelling periodic points). We also discuss Bloch's Principle and provide simple solutions to some problems of Hayman connected with this principle.


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  • 1. Gerardo Aladro and Steven G. Krantz, A criterion for normality in 𝐶ⁿ, J. Math. Anal. Appl. 161 (1991), no. 1, 1–8. MR 1127544, 10.1016/0022-247X(91)90356-5
  • 2. I. N. Baker, Repulsive fixpoints of entire functions, Math. Z. 104 (1968), 252–256. MR 0226009
  • 3. Detlef Bargmann, Simple proofs of some fundamental properties of the Julia set, preprint.
  • 4. G. Barsegian, Geometrical theory of meromorphic functions, manuscript.
  • 5. Walter Bergweiler, On a theorem of Gol'dberg concerning meromorphic solutions of algebraic differential equations, Complex Variables Theory Appl. (to appear).
  • 6. Walter Bergweiler and Alexandre Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995), no. 2, 355–373. MR 1344897, 10.4171/RMI/176
  • 7. Andreas Bolsch, Repulsive periodic points of meromorphic functions, Complex Variables Theory Appl. 31 (1996), no. 1, 75–79. MR 1423240
  • 8. Mario Bonk and Alexandre Eremenko, Schlicht regions for entire and meromorphic functions, preprint.
  • 9. Robert Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213–219. MR 0470252, 10.1090/S0002-9947-1978-0470252-3
  • 10. Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383
  • 11. Huaihui Chen, Yosida functions and Picard values of integral functions and their derivatives, Bull. Austral. Math. Soc. 54 (1996), no. 3, 373–381. MR 1419600, 10.1017/S0004972700021791
  • 12. Huai Hui Chen and Ming Liang Fang, The value distribution of 𝑓ⁿ𝑓’, Sci. China Ser. A 38 (1995), no. 7, 789–798. MR 1360682
  • 13. Huai Hui Chen and Yong Xing Gu, Improvement of Marty’s criterion and its application, Sci. China Ser. A 36 (1993), no. 6, 674–681. MR 1246312
  • 14. Huai Hui Chen and Xin Hou Hua, Normality criterion and singular directions, Proceedings of the Conference on Complex Analysis (Tianjin, 1992) Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA, 1994, pp. 34–40. MR 1343494
  • 15. Chi Tai Chuang, Normal families of meromorphic functions, World Scientific Publishing Co., Inc., River Edge, NJ, 1993. MR 1249270
  • 16. J. Clunie, On a result of Hayman, J. London Math. Soc. 42 (1967), 389–392. MR 0214769
  • 17. J. Clunie and W. K. Hayman, The spherical derivative of integral and meromorphic functions, Comment. Math. Helv. 40 (1966), 117–148. MR 0192055
  • 18. David Drasin, Normal families and the Nevanlinna theory, Acta Math. 122 (1969), 231–263. MR 0249592
  • 19. Alexandre Eremenko, Bloch radius, normal families, and quasiregular mappings, preprint.
  • 20. -, Normal holomorphic curves from parabolic regions to projective spaces, preprint.
  • 21. John Erik Fornæss, Dynamics in several complex variables, CBMS Regional Conference Series in Mathematics, vol. 87, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. MR 1363948
  • 22. Günter Frank and Wilhelm Schwick, A counterexample to the generalized Bloch principle, New Zealand J. Math. 23 (1994), no. 2, 121–123. MR 1313447
  • 23. Günter Frank and Yufei Wang, On the meromorphic solutions of algebraic differential equations, preprint.
  • 24. Hirotaka Fujimoto, On the number of exceptional values of the Gauss maps of minimal surfaces, J. Math. Soc. Japan 40 (1988), no. 2, 235–247. MR 930599, 10.2969/jmsj/04020235
  • 25. Hirotaka Fujimoto, Value distribution theory of the Gauss map of minimal surfaces in 𝑅^{𝑚}, Aspects of Mathematics, E21, Friedr. Vieweg & Sohn, Braunschweig, 1993. MR 1218173
  • 26. A.A. Gol'dberg, On single-valued solutions of first-order differential equations, Ukrain. Math. Zh. 8 (1956), 254-261.
  • 27. W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. (2) 70 (1959), 9–42. MR 0110807
  • 28. W. K. Hayman, Research problems in function theory, The Athlone Press University of London, London, 1967. MR 0217268
  • 29. Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. MR 0201608
  • 30. A. Hinkkanen, Normal families and Ahlfor’s five islands theorem, New Zealand J. Math. 22 (1993), no. 2, 39–41. MR 1244021
  • 31. Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
  • 32. Yung-hsing Ku, Sur les familles normales de fonctions méromorphes, Sci. Sinica 21 (1978), no. 4, 431–445 (French). MR 511294
  • 33. Ku Yongxing, Un critère de normalité des familles de fonctions méromorphes, Sci. Sinica Special Issue 1 (1979), 267-274 (Chinese).
  • 34. Serge Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, New York, 1987. MR 886677
  • 35. Peter Lappan, A criterion for a meromorphic function to be normal, Comment. Math. Helv. 49 (1974), 492–495. MR 0379850
  • 36. Peter Lappan, A uniform approach to normal families, Rev. Roumaine Math. Pures Appl. 39 (1994), no. 7, 691–702. Travaux de la Conférence Internationale d’Analyse Complexe et du 7^{𝑒} Séminaire Roumano-Finlandais. MR 1319185
  • 37. Song-Ying Li and Hui Chun Xie, On normal families of meromorphic functions, Acta Math. Sinica 29 (1986), no. 4, 468–476 (Chinese). MR 867694
  • 38. F. Marty, Recherches sur le répartition des valeurs d'une fonction méromorphe, Ann. Fac. Sci. Univ. Toulouse (3) 23 (1931), 183-261.
  • 39. David Minda, Yosida functions, Lectures on complex analysis (Xian, 1987) World Sci. Publishing, Singapore, 1988, pp. 197–213. MR 996476
  • 40. Ruth Miniowitz, Normal families of quasimeromorphic mappings, Proc. Amer. Math. Soc. 84 (1982), no. 1, 35–43. MR 633273, 10.1090/S0002-9939-1982-0633273-X
  • 41. Carlo Miranda, Sur un nouveau critère de normalité pour les familles de fonctions holomorphes, Bull. Soc. Math. France 63 (1935), 185-196.
  • 42. Paul Montel, Leçons sur les familles normales des fonctions analytiques et leurs applications, Gauthier-Villars, Paris, 1927.
  • 43. Erwin Mues, Über ein Problem von Hayman, Math. Z. 164 (1979), no. 3, 239–259 (German). MR 516609, 10.1007/BF01182271
  • 44. Rolf Nevanlinna, Analytic functions, Translated from the second German edition by Phillip Emig. Die Grundlehren der mathematischen Wissenschaften, Band 162, Springer-Verlag, New York-Berlin, 1970. MR 0279280
  • 45. I. B. Oshkin, On a condition for the normality of families of holomorphic functions, Uspekhi Mat. Nauk 37 (1982), no. 2(224), 221–222 (Russian). MR 650781
  • 46. Robert Osserman, Proof of a conjecture of Nirenberg, Comm. Pure Appl. Math. 12 (1959), 229–232. MR 0105700
  • 47. Robert Osserman, Minimal surfaces in 𝑅³, Global differential geometry, MAA Stud. Math., vol. 27, Math. Assoc. America, Washington, DC, 1989, pp. 73–98. MR 1013809
  • 48. Xue Cheng Pang, Bloch’s principle and normal criterion, Sci. China Ser. A 32 (1989), no. 7, 782–791. MR 1057999
  • 49. Xue Cheng Pang, On normal criterion of meromorphic functions, Sci. China Ser. A 33 (1990), no. 5, 521–527. MR 1070538
  • 50. Pang Xue-cheng and Lawrence Zalcman, On theorems of Hayman and Clunie, New Zealand J. Math. (to appear).
  • 51. -, Normal families and shared values, preprint.
  • 52. Seppo Rickman, On the number of omitted values of entire quasiregular mappings, J. Analyse Math. 37 (1980), 100–117. MR 583633, 10.1007/BF02797681
  • 53. Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941
  • 54. Abraham Robinson, Metamathematical problems, J. Symbolic Logic 38 (1973), 500–516. MR 0337471
  • 55. Antonio Ros, The Gauss map of minimal surfaces, preprint.
  • 56. H. L. Royden, A criterion for the normality of a family of meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 499–500. MR 802513
  • 57. Lee A. Rubel, Four counterexamples to Bloch’s principle, Proc. Amer. Math. Soc. 98 (1986), no. 2, 257–260. MR 854029, 10.1090/S0002-9939-1986-0854029-2
  • 58. Stanis{\l}aw Saks and Antoni Zymund, Analytic Functions, 3rd ed., Elsevier, 1971.
  • 59. Joel L. Schiff, Normal families, Universitext, Springer-Verlag, New York, 1993. MR 1211641
  • 60. Wilhelm Schwick, Normality criteria for families of meromorphic functions, J. Analyse Math. 52 (1989), 241–289. MR 981504, 10.1007/BF02820480
  • 61. Wilhelm Schwick, On a normality criterion of H. L. Royden, New Zealand J. Math. 23 (1994), no. 1, 91–92. MR 1279130
  • 62. Wilhelm Schwick, Repelling periodic points in the Julia set, Bull. London Math. Soc. 29 (1997), no. 3, 314–316. MR 1435565, 10.1112/S0024609396007035
  • 63. Jussi Väisälä, Lectures on 𝑛-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009
  • 64. H. Wu, Some theorems on projective hyperbolicity, J. Math. Soc. Japan 33 (1981), no. 1, 79–104. MR 597482, 10.2969/jmsj/03310079
  • 65. Guo Fen Xue and Xue Cheng Pang, A criterion for normality of a family of meromorphic functions, J. East China Norm. Univ. Natur. Sci. Ed. 2 (1988), 15–22 (Chinese, with English summary). MR 981092
  • 66. Yang Le and Chang Kuang-hou, Recherches sur la normalité des familles de fonctions analytiques à des valeurs multiples, I. Un nouveau critère et quelques applications, Sci. Sinica 14 (1965), 1258-1271.
  • 67. -, Recherches sur la normalité des familles de fonctions analytiques à des valeurs multiples, II. Généralisations, Sci. Sinica 15 (1966), 433-453.
  • 68. Lawrence Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813–817. MR 0379852
  • 69. Lawrence Zalcman, Modern perspectives on classical function theory, Rocky Mountain J. Math. 12 (1982), no. 1, 75–92. MR 649740, 10.1216/RMJ-1982-12-1-75
  • 70. -, Normal families revisited, Complex Analysis and Related Topics (J.J.O.O. Wiegerinck, ed.), University of Amsterdam, 1993, pp. 149-164.
  • 71. -, On some questions of Hayman, unpublished manuscript, 5pp., 1994.
  • 72. -, New light on normal families, Proceedings of the Ashkelon Workshop on Complex Function Theory (May, 1996) (L. Zalcman, ed.), Bar-Ilan Univ., 1997, pp. 237-245. CMP 98:03

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Additional Information

Lawrence Zalcman
Affiliation: Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: zalcman@macs.biu.ac.il

DOI: http://dx.doi.org/10.1090/S0273-0979-98-00755-1
Keywords: Normal families, Picard's Theorem, algebraic differential equations, Julia set, Bloch's Principle
Received by editor(s): October 15, 1997
Received by editor(s) in revised form: May 26, 1998
Article copyright: © Copyright 1998 American Mathematical Society