Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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NORMALITY CRITERIA FOR A FAMILY OF HOLOMORPHIC FUNCTIONS CONCERNING THE TOTAL DERIVATIVE IN SEVERAL COMPLEX VARIABLES

  • Cao, Tingbin (Department of Mathematics Nanchang University) ;
  • Liu, Zhixue (Department of Mathematics Nanchang University)
  • Received : 2015.09.06
  • Published : 2016.11.01
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Abstract

In this paper, we investigate a family of holomorphic functions in several complex variables concerning the total derivative (or called radial derivative), and obtain some well-known normality criteria such as the Miranda's theorem, the Marty's theorem and results on the Hayman's conjectures in several complex variables. A high-dimension version of the famous Zalcman's lemma for normal families is also given.

Keywords

Acknowledgement

Supported by : NSFC, CPSF

References

  1. D. Bargmann, M. Bonk, A. Hinkkanen, and G. J. Martin, Families of meromorphic functions avoiding continuous functions, J. Anal. Math. 79 (1999), 379-387. https://doi.org/10.1007/BF02788248
  2. W. Bergweiler and W. Eremenko, On the singularities of the inverse to a mermorphic function of finite order, Rev. Mat. Iberoam 11 (1995), no. 2, 335-373.
  3. A. Bloch, Sur les systemes de fonctions holomorphes a varietes lineaires lacunaires, Ann. Sci. Ecole Norm. Sup. 43 (1926), no. 3, 309-362. https://doi.org/10.24033/asens.772
  4. H. H. Chen ad M. L. Fang, On value distribution of fnf′, Sci. China Ser. A 38 (1995), 789-798.
  5. Q. Chen, S. Nevo, and X. C. Pang, A general differential inequality of the k-th derivative that leads to normality, Ann. Acad. Sci. Fenn. Math. 38 (2013), no. 2, 691-695. https://doi.org/10.5186/aasfm.2013.3833
  6. D. Drasin, Normal families and the Nevanlinna theory, Acta Math. 122 (1969), 231-263. https://doi.org/10.1007/BF02392012
  7. M. Erwin, Uberein Problem Von Hayman, Math. Z. 8 (1979), no. 1, 239-259.
  8. H. Fujimoto, Extensions of the big Picard's theorem, Tohoku Math J. 24 (1972), 415-422. https://doi.org/10.2748/tmj/1178241480
  9. J. Grahl and S. Nevo, Spherical derivatives and normal families, J. Anal. Math. 117 (2012), 119-128. https://doi.org/10.1007/s11854-012-0016-4
  10. J. Grahl and S. Nevo, An extension of one direction in Marty's normality criterion, Monatsh. Math. 174 (2014), no. 2, 205-217. https://doi.org/10.1007/s00605-013-0561-7
  11. M. Green, Holomorphic maps into complex projective space omitting hyperplanes, Trans. Amer. Math. Soc. 169 (1972), 89-103. https://doi.org/10.1090/S0002-9947-1972-0308433-6
  12. Y. X. Gu, On normal families of meromorphic functions, Sci. China Ser. A (1978), no. 4, 373-384.
  13. W. K. Hayman, Research Problems in Function Theory, London: Athlone Press, 1967.
  14. Z. Hu, Extended Ces'aro operators on mixed norm spaces, Proc. Amer. Math. Soc. 131 (2003), no. 7, 2171-2179. https://doi.org/10.1090/S0002-9939-02-06777-1
  15. L. Jin, Theorem of Picard type for entire functions of several complex variables, Kodai Math. J. 26 (2003), no. 2, 221-229. https://doi.org/10.2996/kmj/1061901063
  16. S. Y. Li, The normality criterion of a class of meromorphic functions, J. Fujian Norm. Univ. 2 (1984), 156-158.
  17. S. Y. Li and C. H. Xie, On normal families of meromorphic functions, Acta Math. Sin. 4 (1986), 468-476.
  18. B. Li and C. Ouyang, Higher radial derivative of functions of $Q_p$ spaces and its applications, J. Math. Anal. Appl. 327 (2007), no. 2, 1257-1272. https://doi.org/10.1016/j.jmaa.2006.04.088
  19. X. J. Liu, S. Nevo, and X. C. Pang, Differential inequalities, normality and quasi-normality, Acta Math. Sin. (Engl. Ser.) 30 (2014), no. 2, 277-282. https://doi.org/10.1007/s10114-014-2542-8
  20. F. Lu, Theorems of Picard type for meromorphic function of several complex variables, Complex Var. Elliptic Equ. 58 (2013), no. 8, 1085-1092. https://doi.org/10.1080/17476933.2011.627440
  21. F. Marty, Recherches sur la repartition des valeurs dune fonction meromorphe, Ann. Fac. Sci. Univ. Toulouse Sci. Math. Sci. Phys. (3) 23 (1931), 183-261.
  22. C. Miranda, Sur un nouveau critere de normalite pour les familles des fonctions holomorphes, Bull. Sci. Math. France 63 (1935), 185-196.
  23. P. Montel, Lecons sur les families normales de fonctions analytiques et leur applicaeions, Coll. Borel, 1927.
  24. E. Nochka, On the theory of meromorphic functions, Soviet Math. Dokl. 27 (1983), 377-381.
  25. I. Oshkin, A normal criterion of families of holomorphic functions, (Russian) Usp. Mat. Nauk. 37 (1982), no. 2, 221-222.
  26. X. C. Pang, Bloch's principle and normal criterion, Sci. China Ser. A 32 (1989), no. 7, 782-791.
  27. X. C. Pang, On normal criterion of meromorphic functions, Sci. China Ser. A 33 (1990), no. 5, 521-527.
  28. 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.
  29. W. Rudin, Function Theory in the Unit Ball of $\mathbb{C}^n$, Springer-Verlag, New York-Berlin, 1980.
  30. J. Schiff, Normal Families, Springer, New York, 1993.
  31. Z. H. Tu, Normality criteria for families of holomorphic mappings of several complex variables into $P^n(\mathbb{C})$, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1039-1049. https://doi.org/10.1090/S0002-9939-99-04610-9
  32. Z. H. Tu and S. S. Zhang, Normal families of holomorphic mappings of several complex variables into $\mathbb{L}^1(\mathbb{C})$, Acta. Math. Sin. (Chin. Ser.) 53 (2010), no. 6, 1045-1050.
  33. L. Yang and G. H. Zhang, Recherches sur la normalite des familiesn de fonctions analytiques ades valeurs multiples. I. Un nouveau critere at quelques applications, Sci. Sinica 14 (1965), 1258-1271
  34. L. Yang and G. H. Zhang, Recherches sur la normalite des familiesn de fonctions analytiques ades valeurs multiples. II. Generalisations, Sci. Sinica 15 (1996), 433-453.
  35. Y. S. Ye, A new normal criterion and its application, Chin. Ann. Math. Ser. A(Supplement) 12 (1991), 44-49.
  36. L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813-817. https://doi.org/10.2307/2319796
  37. L. Zalcman, On some quesitions of Hayman, unpublished manuscript, 5 pp., 1994.
  38. K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Graduate Text in Mathematics 226, Springer, New York, 2005.