On one problem of uniqueness of meromorphic functions concerning small functions

Author:
Hong-Xun Yi

Journal:
Proc. Amer. Math. Soc. **130** (2002), 1689-1697

MSC (2000):
Primary 30D35; Secondary 30D30

Published electronically:
October 17, 2001

MathSciNet review:
1887016

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we show that if two non-constant meromorphic functions and satisfy for , where are five distinct small functions with respect to and , and is a positive integer or with , then . As a special case this also answers the long-standing problem on uniqueness of meromorphic functions concerning small functions.

**1.**W. K. Hayman,*Meromorphic functions*, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR**0164038****2.**Katsuya Ishizaki and Nobushige Toda,*Unicity theorems for meromorphic functions sharing four small functions*, Kodai Math. J.**21**(1998), no. 3, 350–371. MR**1664754**, 10.2996/kmj/1138043945**3.**Bao Qin Li,*Uniqueness of entire functions sharing four small functions*, Amer. J. Math.**119**(1997), no. 4, 841–858. MR**1465071****4.**Yu Hua Li,*Entire functions that share four functions IM*, Acta Math. Sinica (Chin. Ser.)**41**(1998), no. 2, 249–260 (Chinese, with English and Chinese summaries). MR**1656489****5.**Y. H. Li,*Meromorphic functions which share four or five small functions*, J. Math. Res. Exp.**20**(2000), 94-96.**6.**Yuhua Li and Jianyong Qiao,*The uniqueness of meromorphic functions concerning small functions*, Sci. China Ser. A**43**(2000), no. 6, 581–590. MR**1775265**, 10.1007/BF02908769**7.**Rolf Nevanlinna,*Le théorème de Picard-Borel et la théorie des fonctions méromorphes*, Chelsea Publishing Co., New York, 1974 (French). Reprinting of the 1929 original. MR**0417418****8.**Manabu Shirosaki,*An extension of unicity theorem for meromorphic functions*, Tohoku Math. J. (2)**45**(1993), no. 4, 491–497. MR**1245716**, 10.2748/tmj/1178225843**9.**N. Toda,*Some generalizations of the unicity theorem of Nevanlinna*, Proc. Japan Acad. Ser. A Math. Sci.**69**(1993), 61-65. MR**94d:3D056****10.**H. X. Yi and C. C. Yang,*Uniqueness Theory of Meromorphic Functions*, Pure and Applied Math. Monographs No. 32, Science Press, Beijing, 1995.**11.**Qing De Zhang,*A uniqueness theorem for meromorphic functions with respect to slowly growing functions*, Acta Math. Sinica**36**(1993), no. 6, 826–833 (Chinese, with Chinese summary). MR**1270289**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
30D35,
30D30

Retrieve articles in all journals with MSC (2000): 30D35, 30D30

Additional Information

**Hong-Xun Yi**

Affiliation:
Department of Mathematics, Shandong University, Jinan 250100, People’s Republic of China

Email:
hxyi@sdu.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-01-06245-1

Keywords:
Meromorphic function,
small function,
uniqueness theorem

Received by editor(s):
September 22, 2000

Received by editor(s) in revised form:
December 1, 2000

Published electronically:
October 17, 2001

Additional Notes:
This work was supported by the NSFC (NO. 19871050) and the RFDP (No. 98042209).

Communicated by:
Juha M. Heinonen

Article copyright:
© Copyright 2001
American Mathematical Society