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http://dx.doi.org/10.4134/JKMS.j150546

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)
Publication Information
Journal of the Korean Mathematical Society / v.53, no.6, 2016 , pp. 1391-1409 More about this Journal
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
holomorphic functions; several complex variables; normal family; total derivative;
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1 E. Nochka, On the theory of meromorphic functions, Soviet Math. Dokl. 27 (1983), 377-381.
2 I. Oshkin, A normal criterion of families of holomorphic functions, (Russian) Usp. Mat. Nauk. 37 (1982), no. 2, 221-222.
3 X. C. Pang, Bloch's principle and normal criterion, Sci. China Ser. A 32 (1989), no. 7, 782-791.
4 X. C. Pang, On normal criterion of meromorphic functions, Sci. China Ser. A 33 (1990), no. 5, 521-527.
5 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.
6 W. Rudin, Function Theory in the Unit Ball of $\mathbb{C}^n$, Springer-Verlag, New York-Berlin, 1980.
7 J. Schiff, Normal Families, Springer, New York, 1993.
8 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.   DOI
9 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.
10 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
11 Y. S. Ye, A new normal criterion and its application, Chin. Ann. Math. Ser. A(Supplement) 12 (1991), 44-49.
12 L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813-817.   DOI
13 L. Zalcman, On some quesitions of Hayman, unpublished manuscript, 5 pp., 1994.
14 H. H. Chen ad M. L. Fang, On value distribution of fnf′, Sci. China Ser. A 38 (1995), 789-798.
15 K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Graduate Text in Mathematics 226, Springer, New York, 2005.
16 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.
17 D. Bargmann, M. Bonk, A. Hinkkanen, and G. J. Martin, Families of meromorphic functions avoiding continuous functions, J. Anal. Math. 79 (1999), 379-387.   DOI
18 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.
19 A. Bloch, Sur les systemes de fonctions holomorphes a varietes lineaires lacunaires, Ann. Sci. Ecole Norm. Sup. 43 (1926), no. 3, 309-362.   DOI
20 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.   DOI
21 D. Drasin, Normal families and the Nevanlinna theory, Acta Math. 122 (1969), 231-263.   DOI
22 M. Erwin, Uberein Problem Von Hayman, Math. Z. 8 (1979), no. 1, 239-259.
23 H. Fujimoto, Extensions of the big Picard's theorem, Tohoku Math J. 24 (1972), 415-422.   DOI
24 J. Grahl and S. Nevo, Spherical derivatives and normal families, J. Anal. Math. 117 (2012), 119-128.   DOI
25 J. Grahl and S. Nevo, An extension of one direction in Marty's normality criterion, Monatsh. Math. 174 (2014), no. 2, 205-217.   DOI
26 Z. Hu, Extended Ces'aro operators on mixed norm spaces, Proc. Amer. Math. Soc. 131 (2003), no. 7, 2171-2179.   DOI
27 M. Green, Holomorphic maps into complex projective space omitting hyperplanes, Trans. Amer. Math. Soc. 169 (1972), 89-103.   DOI
28 Y. X. Gu, On normal families of meromorphic functions, Sci. China Ser. A (1978), no. 4, 373-384.
29 W. K. Hayman, Research Problems in Function Theory, London: Athlone Press, 1967.
30 L. Jin, Theorem of Picard type for entire functions of several complex variables, Kodai Math. J. 26 (2003), no. 2, 221-229.   DOI
31 S. Y. Li, The normality criterion of a class of meromorphic functions, J. Fujian Norm. Univ. 2 (1984), 156-158.
32 S. Y. Li and C. H. Xie, On normal families of meromorphic functions, Acta Math. Sin. 4 (1986), 468-476.
33 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.   DOI
34 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.   DOI
35 F. Lu, Theorems of Picard type for meromorphic function of several complex variables, Complex Var. Elliptic Equ. 58 (2013), no. 8, 1085-1092.   DOI
36 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.
37 C. Miranda, Sur un nouveau critere de normalite pour les familles des fonctions holomorphes, Bull. Sci. Math. France 63 (1935), 185-196.
38 P. Montel, Lecons sur les families normales de fonctions analytiques et leur applicaeions, Coll. Borel, 1927.