Borel directions and the uniqueness of algebroid functions. (English) Zbl 1494.30058

Tan, Yang; Kong, Yinying

Rocky Mt. J. Math. 52, No. 3, 1063-1072 (2022).

Summary: In this paper, using Nevanlinna theory we discuss the relations between the Borel directions and the uniqueness of algebroid functions. We get several uniqueness theorems of algebroid functions in angular domain which contains the Borel directions. These results extend those for meromorphic functions obtained by some scholars.

MSC:

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

Keywords:

algebroid functions; Borel directions; uniqueness

References:

[1]	Y. He, “A uniqueness theorem of algebroid functions and systems of holomorphic functions”, Pure Appl. Math. 1 (1985), 24-31. · Zbl 1070.30502
[2]	Y. Z. He and X. Z. Xiao, “Algebroid functions and ordinary differential equations in the complex domain”, Science Press, Beijing (1988).
[3]	K.-L. Hiong, Sur les fonctions entières et les fonctions méromorphes d’ordre infini, NUMDAM, 1934.
[4]	H. Liu and D. Sun, “Uniqueness theorem of algebroidal functions in the unit disc”, Northeast. Math. J. 24:6 (2008), 511-520. · Zbl 1199.30206
[5]	H. Liu, D. Sun, and Z. Mao, “Meromorphic functions in the unit disc that share slowly growing functions in an angular domain”, Comput. Math. Appl. 62:12 (2011), 4539-4546. · Zbl 1236.30029 · doi:10.1016/j.camwa.2011.10.033
[6]	J. R. Long and P. C. Wu, “Borel directions and uniqueness of meromorphic functions”, Chinese Ann. Math. Ser. A 33:3 (2012), 261-266. · Zbl 1274.30123
[7]	Z. Mao and H. Liu, “Meromorphic functions in the unit disk that share values in an angular domain”, J. Math. Anal. Appl. 359:2 (2009), 444-450. · Zbl 1169.30009 · doi:10.1016/j.jmaa.2009.05.043
[8]	D. Sun and Z. Gao, “On the operations of algebroid functions”, Acta Math. Sci. Ser. B (Engl. Ed.) 30:1 (2010), 247-256. · Zbl 1224.30168 · doi:10.1016/S0252-9602(10)60042-2
[9]	N. Toda, “Sur les directions de Julia et de Borel des fonctions algebroides”, Nagoya Math. J. 34 (1969), 1-23. · Zbl 0174.12002 · doi:10.1017/S0027763000024429
[10]	C.-C. Yang and H.-X. Yi, “Meromorphic functions sharing sets”, pp. 454-532 in Uniqueness Theory of Meromorphic Functions, Springer, 2003. · doi:10.1007/978-94-017-3626-8_10
[11]	Q. Zhang, “Meromorphic functions sharing values in an angular domain”, J. Math. Anal. Appl. 349:1 (2009), 100-112. · Zbl 1161.30023 · doi:10.1016/j.jmaa.2008.08.014
[12]	J.-H. Zheng, “On uniqueness of meromorphic functions with shared values in one angular domain”, Complex Var. Theory Appl. 48:9 (2003), 777-785. · Zbl 1041.30009 · doi:10.1080/02781070310001599368
[13]	J.-H. Zheng, “On uniqueness of meromorphic functions with shared values in some angular domains”, Canad. Math. Bull. 47:1 (2004), 152-160. · Zbl 1045.30019 · doi:10.4153/CMB-2004-016-1