| Title:
|
On unicity of meromorphic functions due to a result of Yang - Hua (English) |
| Author:
|
Bai, Xiao-Tian |
| Author:
|
Han, Qi |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
43 |
| Issue:
|
2 |
| Year:
|
2007 |
| Pages:
|
93-103 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper studies the unicity of meromorphic(resp. entire) functions of the form $f^nf^{\prime }$ and obtains the following main result: Let $f$ and $g$ be two non-constant meromorphic (resp. entire) functions, and let $a\in \mathbb {C}\backslash \lbrace 0\rbrace $ be a non-zero finite value. Then, the condition that $E_{3)}(a,f^nf^{\prime })=E_{3)}(a,g^ng^{\prime })$ implies that either $f=dg$ for some $(n+1)$-th root of unity $d$, or $f=c_1e^{cz}$ and $g=c_2e^{-cz}$ for three non-zero constants $c$, $c_1$ and $c_2$ with $(c_1c_2)^{n+1}c^2=-a^2$ provided that $n\ge 11$ (resp. $n\ge 6$). It improves a result of C. C. Yang and X. H. Hua. Also, some other related problems are discussed. (English) |
| Keyword:
|
entire functions |
| Keyword:
|
meromorphic functions |
| Keyword:
|
value sharing |
| Keyword:
|
unicity |
| MSC:
|
30D35 |
| idZBL:
|
Zbl 1164.30021 |
| idMR:
|
MR2336962 |
| . |
| Date available:
|
2008-06-06T22:50:44Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/108055 |
| . |
| Reference:
|
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| Reference:
|
[2] Fang M. L., Qiu H. L.: Meromorphic functions that share fixed-points.J. Math. Anal. Appl. 268 (2000), 426–439. Zbl 1030.30028, MR 1896207 |
| Reference:
|
[3] Hayman W. K.: Meromorphic Functions.Clarendon Press, Oxford, 1964. Zbl 0115.06203, MR 0164038 |
| Reference:
|
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| Reference:
|
[5] Hayman W. K.: Picard values of meromorphic functions and their derivatives.Ann. of Math. 70 (1959), 9–42. Zbl 0088.28505, MR 0110807 |
| Reference:
|
[6] Yang C. C.: On deficiencies of differential polynomials II.Math. Z. 125 (1972), 107–112. Zbl 0217.38402, MR 0294642 |
| Reference:
|
[7] Yang C. C., Hua X. H.: Uniqueness and value-sharing of meromorphic functions.Ann. Acad. Sci. Fenn. Math. 22 (1997), 395–406. Zbl 0890.30019, MR 1469799 |
| Reference:
|
[8] Yang C. C., Yi H. X.: Uniqueness Theory of Meromorphic Functions.Science Press & Kluwer Academic Punlishers, Beijing & Dordrecht, 2003. Zbl 1070.30011, MR 2105668 |
| Reference:
|
[9] Yi H. X.: Uniqueness of meromorphic functions and a question of C. C. Yang.Complex Variables Theory Appl. 14 (1990), 169–176. Zbl 0701.30025, MR 1048716 |
| . |
