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Value distribution theory and the research of Yang Lo

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Abstract

Value distribution theory is concerned with the position and frequency of solutions of the equation f(z) = a. Here f may be entire, i.e. an everywhere convergent power series or meromorphic, i.e. the ratio of two such series, or a function in some other domains, such as an angle or a disk.

Yang Lo’s significant contributions to this area will be highlighted. Some of his important contributions to normal families will also be described.

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References

  1. Borel E. Sur les zéros des fonctions entières. Acta Math, 1897, 20: 357–396

    Article  MATH  MathSciNet  Google Scholar 

  2. Drasin D. Normal families and the Nevanlinna theory. Acta Math, 1969, 122: 231–363

    Article  MATH  MathSciNet  Google Scholar 

  3. Drasin D. A meromorphic function with assigned Nevanlinna deficiencies. In: Proceedings of the Sysposium on Complex Analysis. LMS Lecture Notes Series No. 12, 1974, 312–342

  4. Drasin D, Weitsman A. On the Julia directions and Borel directions of entire functions. Proc London Math Soc, 1976, 32: 199–212

    Article  MATH  MathSciNet  Google Scholar 

  5. Gol’dberg A A. Existence and some properties of meromorphic functions for which the Yang Le deficiency relation reaches equality. Mat Stud, 1994, 3: 53–60

    MATH  MathSciNet  Google Scholar 

  6. Hadamard J. Essai sur l’étude des fonctions données par leur développement de Taylor. J Math Pure Appl, 1892, 8: 101–186

    Google Scholar 

  7. Hayman W K. Picard values of meromorphic functions and their derivatives. Ann of Math, 1959, 70: 9–52

    Article  MathSciNet  Google Scholar 

  8. Hayman W K. Research Problems in Function Theory. London: Athlone Press, 1967

    MATH  Google Scholar 

  9. Jensen J L W V. Sur un nouvel et important théorème de la théorie des fonctions. Acta Math, 1899, 22: 359–364

    Article  MATH  MathSciNet  Google Scholar 

  10. Milloux H. Les Fonctions Méromorphes et Leurs Derivées. Paris: Hermann, 1940

    MATH  Google Scholar 

  11. Montel P. Leçons sur les Familles Normales de Fonctions Analytiques et Leurs Applications. Paris: Gauthier-Villars, 1927

    MATH  Google Scholar 

  12. Mues E. Über eine Defekt und Verzweigungsrelation für die Ableitung Meromorpher Funktionen. Manuscripta Math, 1971, 5: 275–2

    Article  MATH  MathSciNet  Google Scholar 

  13. Nevanlinna R. Le Théorème de Picard-Borel et la Théorie des Fonctions Méromorphes. Paris: Gauthier-Villars, 1929

    MATH  Google Scholar 

  14. Oshkin I B. On a test of the normality of holomorphic functions. Uspehi Mat Nauk, 1982, 37: 221–222 (Russian Math Surveys, 1982, 37: 237–238)

    MATH  MathSciNet  Google Scholar 

  15. Pang X, Zalcman L. Normal families and shared values. Bull London Math Soc, 2000, 32: 325–331

    Article  MATH  MathSciNet  Google Scholar 

  16. Picard É. Sur une propriété des fonctions entières. C R Math Acad Sci Paris, 1897, 88: 1024–1027

    Google Scholar 

  17. Zalcman L. A heuristic principle in complex function theory. Amer Math Monthly, 1975, 82: 813–817

    Article  MATH  MathSciNet  Google Scholar 

  18. Zhang G H. On the research of common Borel directions of a meromorphic function with its derivatives. Acta Math Sinica, 1977, 20: 237–247

    MATH  MathSciNet  Google Scholar 

List of Publications (Yang Lo) I. Papers

  1. Yang L. Multiple values of meromorphic functions and of their combinations (in Chinese). Acta Math Sinica, 1964, 14: 428–437

    MATH  Google Scholar 

  2. Yang L. Sur les valeurs quasi-exceptionelles des fonctions holomorphes. Scientia Sinica, 1964, 13: 879–885

    MATH  MathSciNet  Google Scholar 

  3. Yang L, Zhang G H. Recherches sur la normalité des familles de fonctions analytiques à des valeurs multiples, I. Un noveau critère et quelques applications. Scientia Sinica, 1965, 14: 1258–1271

    MATH  MathSciNet  Google Scholar 

  4. Yang L, Shiao S Z. Sur les points de Borel des fonctions méromorphes et de leur dérivées. Scientia Sinica, 1965, 14: 1556–1573

    MATH  MathSciNet  Google Scholar 

  5. Yang L, Zhang G H. Recherches sur la normalité des familles de fonctions analytiques à des valeurs multiples, II. Généralisations. Scientia Sinica, 1966, 15: 433–453

    MATH  MathSciNet  Google Scholar 

  6. Yang L, Zhang G H. Sur la distribution des directions de Borel des functions méromorphes. Scientia Sinica, 1973, 16: 465–482

    MATH  MathSciNet  Google Scholar 

  7. Yang L, Zhang G H. Deficient values and Borel directions of meromorphic functions (in Chinese). Bull Science, 1974, 19: 469–470

    Google Scholar 

  8. Yang L, Zhang G H. Tangential cluster sets and boundary properties of meromorphic functions (in Chinese). Acta Math Sinica, 1975, 18: 136–148

    MATH  Google Scholar 

  9. Yang L, Zhang G H. Deficient values of entire functions (in Chinese). Acta Math Sinica, 1975, 18: 35–53

    MATH  MathSciNet  Google Scholar 

  10. Yang L, Zhang G H. Recherches sur le nombre des valeurs déficientes et le nombre des directions de Borel des fonctions méromorphes. Scientia Sinica, 1975, 18: 23–37

    Google Scholar 

  11. Yang L, Zhang G H. Distribution of Borel directions of entire functions (in Chinese). Acta Math Sinica, 1976, 19: 157–168

    MATH  MathSciNet  Google Scholar 

  12. Yang L, Zhang G H. Sur la construction des fonctions méromorphes ayant des directions singulières données. Scientia Sinica, 1976, 19: 445–459

    MATH  MathSciNet  Google Scholar 

  13. Yang L, Zhang G H. Development of value distribution theory of meromorphic functions (in Chinese). Bull Science, 1977, 22: 375–380

    MATH  Google Scholar 

  14. Yang L, Zhang G H. Deficient values of extremal entire functions. Scientia Sinica, 1977, 20: 421–435

    MATH  MathSciNet  Google Scholar 

  15. Ynag L. Borel directions of meromorphic functions in an angle (in Chinese). Scientia Sinica, 1979, Special Issue (I): 149–164

  16. Yang L, Zhang G H. Deficient values and asymptotic values of entire functions. Scientia Sinica, 1979, Special Issue (II): 190–203

    Google Scholar 

  17. Yang L. Common Borel directions of a meromorphic function and its derivatives. Scientia Sinica, 1979, Special Issue (II): 91–104

    Google Scholar 

  18. Yang L, Hayman W K. Value distribution of holomorphic functions in an angle (in Chinese). Bull Science, 1980, 25: 385–388

    MathSciNet  Google Scholar 

  19. Yang L. Angular distribution and multiple values between entire functions and their derivatives. Scientia Sinica, 1980, 23: 16–39

    MATH  MathSciNet  Google Scholar 

  20. Drasin D, Weitsman A, Yang L, Zhang G H. Deficient values of entire functions and their derivatives. Proc Amer Math Soc, 1981, 82: 607–612

    Article  MATH  MathSciNet  Google Scholar 

  21. Yang L. Deficient functions of meromorphic functions. Scientia Sinica, 1981, 24: 1179–1189

    MATH  MathSciNet  Google Scholar 

  22. Yang L, Zhang G H. Deficient values of meromorphic functions and their derivatives (in Chinese). Acta Math Sinica, 1982, 25: 617–625

    MATH  MathSciNet  Google Scholar 

  23. Yang L. Value distribution of meromorphic functions and their derivatives. Scientia Sinica Ser A, 1982, 25: 572–582

    MATH  Google Scholar 

  24. Yang L, Zhang G H. A general theorem on the number of deficient values of entire functions (in Chinese). Acta Math Sinica, 1982, 25: 244–247

    MATH  MathSciNet  Google Scholar 

  25. Yang L, Zhang G H. Distribution of zeros and poles of meromorphic functions and their filling disks. Scientia Sinica Ser A, 1982, 25: 371–383

    Google Scholar 

  26. Hayman W K, Yang L. Growth and values of functions regular in an angle. Proc London Math Soc, 1982, 44: 193–214

    Article  MATH  MathSciNet  Google Scholar 

  27. Yang L. Meromorphic functions and their derivatives. J London Math Soc, 1982, 25: 288–296

    Article  MATH  MathSciNet  Google Scholar 

  28. Yang L. Bloch principle and related topics (in Chinese). J East China Normal Univ, 1983, 1–6

  29. Yang L. A fundamental inequality and its applications. Chin Ann Math Ser B, 1983, 4: 339–346

    Google Scholar 

  30. Yang L. Normal families and differential polynomials. Scientia Sinica Ser A, 1983, 26: 673–686

    MATH  Google Scholar 

  31. Lang R L, Yang L. BMO functions in spaces of homogeneous type. Scientia Sinica Ser A, 1984, 27: 695–708

    Google Scholar 

  32. Yang L. Quasi-deficient values of meromorphic functions (in Chinese). Acta Math Sinica, 1984, 27: 249–256

    MATH  MathSciNet  Google Scholar 

  33. Yang L, Zhang Q D. New singular directions of meromorphic functions. Scientia Sinica Ser A, 1984, 27: 352–366

    MATH  Google Scholar 

  34. Yang L. A general criterion for normality. Acta Math Sinica New Ser, 1985, 1: 181–192

    Article  MATH  Google Scholar 

  35. Yang L, Chuang C T. Distribution of the values of meromorphic functions. Contemp Math, 1985, 48: 21–63

    MathSciNet  Google Scholar 

  36. Yang L, Zhang Q D. Normality for families of meromorphic functions. Scientia Sinica Ser A, 1986, 29: 1263–1274

    MATH  Google Scholar 

  37. Yang L. Angular distribution of meromorphic functions and their derivatives (in Chinese). Acta Math Sinica, 1986, 29: 355–361

    MATH  MathSciNet  Google Scholar 

  38. Yang L. Some results on value distribution theory of entire and meromorphic functions. Adv Sci China Math, 1986, 1: 199–223

    Google Scholar 

  39. Yang L. Normal families and fix-points of meromorphic functions. Indiana Univ Math J, 1986, 35: 179–191

    Article  MATH  MathSciNet  Google Scholar 

  40. Yang L. On the deficient values and angular distribution (in Chinese). Bull Science, 1987, 32: 885–888

    Google Scholar 

  41. Yang L. Angular distribution of meromorphic functions in the unit disk. In: Complex Analysis. Basel: Birkhäuser-Verlag, 1988, 239–245

    Google Scholar 

  42. Yang L. Deficient values and angular distribution of entire functions. Trans Amer Math Soc, 1988, 308: 583–601

    Article  MATH  MathSciNet  Google Scholar 

  43. Yang L. Growth and angular distribution of entire functions. Complex Var, 1989, 13: 155–160

    MATH  Google Scholar 

  44. Yang L. Precise estimate of total deficiency for derivatives of meromorphic functions (in Chinese). Bull Science, 1990, 35: 1208–1210

    Google Scholar 

  45. Yang L. Results and problems in the value distribution theory of meromorphic functions (in Chinese). In: Proceedings of Conference in Memory of T.H. Min. Jinan: Shandong University Press, 1990, 49–55

    Google Scholar 

  46. Yang L. Precise estimate of total deficiency of meromorphic derivatives. J Anal Math, 1990, 55: 287–296

    Article  MATH  Google Scholar 

  47. Yang L. Deficient values and deficient functions of meromorphic functions. In: International Symposium in Memory of Hua Loo Keng. Berlin-Beijing: Springer-Verlag & Science Press, 1991, 313–321

    Google Scholar 

  48. Yang L. On the total deficiency of meromorphic functions concerning their derivatives (in Chinese). Bull Science, 1991, 36: 1761–1763

    Google Scholar 

  49. Yang L. Precise fundamental inequalities and sum of deficiencies. Sci China Ser A, 1991, 34: 157–165

    MATH  MathSciNet  Google Scholar 

  50. Yang L. Some recent results and problems in the theory of value distribution. In: Proceedings of Symposium on Value Distribution Theory in Several Complex Variables. Notre Dame: University of Notre Dame Press, 1992, 157–171

    Google Scholar 

  51. Yang L. Total deficiency of meromorphic functions and their derivatives. In: Proceedings of the Asian Math Conference. Singapore: World Scitific Publication Company, 1992, 519-523

    Google Scholar 

  52. Yang L, Wang Y F. Drasin’s problem and Mues’ conjecture. Sci China Ser A, 1992, 35: 1180–1190

    MATH  MathSciNet  Google Scholar 

  53. Yang L, Wang Y F, Yang C C. On the zeros of (f k)(n) f − a (in Chinese). Bull Science, 1993, 38: 2209–2212

    Google Scholar 

  54. Yang L. Several results and problems in the theory of value distribution. SEA Bull Math, 1993, Special Issue: 175–183

  55. Yang L. Angular distribution of values of ff′. Sci China Ser A, 1994, 27: 284–294

    Google Scholar 

  56. Yang L. Kinglai Hiong: A brief outline of his life and works. In: Proceedings of the International Conference on Complex Ananlysis. Somerville, MA: International Press, 1994, 3–7

    Google Scholar 

  57. Sun D C, Yang L. Value distribution of quasiconformal mappings. Complex Var, 1997, 34: 219–229

    MATH  MathSciNet  Google Scholar 

  58. Yang L. History and development of value distribution theory in China. Complex Var Theory Appl, 1998, 37: 507–523

    MATH  Google Scholar 

  59. Yang L. Some results and problems in the theory of value distribution. In: Proceedings of the Second Asian Math Conference. Singapore: World Scientific Publication Company, 1998, 85–93

    Google Scholar 

  60. Sun D C, Yang L. Quasi-rational dynamics (in Chinese). Bull Science, 1999, 44: 1481–1483

    Google Scholar 

  61. Sun D C, Yang L. Quasirational dynamical system (in Chinese). Chin Ann Math Ser A, 1999, 20: 673–684

    MATH  MathSciNet  Google Scholar 

  62. Yang L. In memory of Professor C.T. Chuang. In: Proceedings of the Eighth International Colloquium on Finite or Infinite Dimensional Complex Analysis. Jinan: Shangdong Sci & Tech Press, 2001, 1–4

    Google Scholar 

  63. Sun D C, Yang L. Iteration of quasi-rational mapping. Progr Natur Sci (English Ed), 2001, 11: 16–25

    MathSciNet  Google Scholar 

  64. Sun D C, Yang L. Normal family of algebroidal functions. Sci China Ser A, 2001, 44: 1271–1277

    Article  MATH  MathSciNet  Google Scholar 

  65. Sun D C, Yang L. Normal family of quasimeromorphic mappings. Sci China Ser A, 2003, 46: 440–449

    Article  MathSciNet  Google Scholar 

II. Monographes

  1. Yang L. Theory of Value-Distribution and its New Development (in Chinese). Beijing: Science Press, 1982

    Google Scholar 

  2. Yang L. Value Distribution Theory. Berlin: Springer-Verlag, 1993

    MATH  Google Scholar 

III. Proceedings

  1. International Smyposium in Memory of Hua Loo Keng, vol. I, Number Theory. Berlin: Springer-Verlag, 1991 (edited with Sheng Gong, Qi-Keng Lu and Yuan Wang)

  2. International Smyposium in Memory of Hua Loo Keng, vol. II, Analysis. Berlin: Springer-Verlag, 1991 (edited with Sheng Gong, Qi-Keng Lu and Yuan Wang)

  3. Proceedings of Asian Mathematical Conference 1990. Singapore: World Scientific Publication Company, 1992 (edited with Zhong Li, K.P. Shum and C.C. Yang)

  4. Lectures Notes in Contemporary Mathematics, vol. 2. Beijing-New York: Science Press, 1992 (edited with Bingren Li and Shixue Zhu)

  5. Proceedings of International Conference on Complex Analysis, Nankai Institute of Mathematics, 1992. Somerville: International Press, 1994 (edited with Zhong Li, Fuyao Ren and Shunyan Zhang)

  6. Sixty years of Chinese Mathematical Society (in Chinese). Changsha: Hunan Education Press, 1996

  7. First International Congress of Chinese Mathematicians, Proceedings of ICCM 98. Providence, RI-Somerville, MA: American Mathematical Society & International Press (edited with Shing-Tong Yau)

  8. Proceedings of the Eighth International Colloquium on Finite or Infinite Dimensional Complex Analysis. Jinan: Shandong Sci & Tech Press, 2001 (edited with Hong-Xun Yi)

  9. Proceedings of Third Asian Mathematics Conference 2000. Singapore: World Scitific Publication Company, 2002 (edited with T. Sunada & P.W. Sy)

  10. Second International Congress of Chinese Mathematicians, Proceedings of ICCM 2001. Somerville, MA: International Press, 2004 (edited with Chang-Shou Lin & Shing-Tong Yau)

  11. Proceedings of the 4th International Congress of Chinese Mathematicians, vols. I, II, III. Beijing: Higher Education Press, 2007 (edited with Lizhen Ji, Kefeng Liu and Shing-Tong Yau)

  12. Proceedings of the 13th International Conference on Finite and Infinite Dimensional Complex Analysis and Applications. Singapore: World Scitific Publication Company, 2006 (edited with Yuefei Wang, Shengjian Wu and Hasi Wulan)

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Correspondence to ShengJian Wu.

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Dedicated to Professor Yang Lo on the Occasion of his 70th Birthday

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Hayman, W.K., Wu, S. Value distribution theory and the research of Yang Lo. Sci. China Math. 53, 513–522 (2010). https://doi.org/10.1007/s11425-010-0045-3

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