On unicity of meromorphic functions due to a result of Yang - Hua

Xiao-Tian Bai; Qi Han

Archivum Mathematicum (2007)

  • Volume: 043, Issue: 2, page 93-103
  • ISSN: 0044-8753

Abstract

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This paper studies the unicity of meromorphic(resp. entire) functions of the form f n f ' and obtains the following main result: Let f and g be two non-constant meromorphic (resp. entire) functions, and let a ∈ ℂ ∖ { 0 } be a non-zero finite value. Then, the condition that E 3 ) ( a , f n f ' ) = E 3 ) ( a , g n g ' ) implies that either f = d g for some ( n + 1 ) -th root of unity d , or f = c 1 e c z and g = c 2 e - c z for three non-zero constants c , c 1 and c 2 with ( c 1 c 2 ) n + 1 c 2 = - a 2 provided that n ≥ 11 (resp. n ≥ 6 ). It improves a result of C. C. Yang and X. H. Hua. Also, some other related problems are discussed.

How to cite

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Bai, Xiao-Tian, and Han, Qi. "On unicity of meromorphic functions due to a result of Yang - Hua." Archivum Mathematicum 043.2 (2007): 93-103. <http://eudml.org/doc/250164>.

@article{Bai2007,
abstract = {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.},
author = {Bai, Xiao-Tian, Han, Qi},
journal = {Archivum Mathematicum},
keywords = {entire functions; meromorphic functions; value sharing; unicity; entire functions; meromorphic functions; value sharing; unicity},
language = {eng},
number = {2},
pages = {93-103},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On unicity of meromorphic functions due to a result of Yang - Hua},
url = {http://eudml.org/doc/250164},
volume = {043},
year = {2007},
}

TY - JOUR
AU - Bai, Xiao-Tian
AU - Han, Qi
TI - On unicity of meromorphic functions due to a result of Yang - Hua
JO - Archivum Mathematicum
PY - 2007
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 043
IS - 2
SP - 93
EP - 103
AB - 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.
LA - eng
KW - entire functions; meromorphic functions; value sharing; unicity; entire functions; meromorphic functions; value sharing; unicity
UR - http://eudml.org/doc/250164
ER -

References

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  4. Hayman W. K., Research Problems in Function Theory, Athlore Press (Univ. of London), 1967. (1967) Zbl0158.06301MR0217268
  5. Hayman W. K., Picard values of meromorphic functions and their derivatives, Ann. of Math. 70 (1959), 9–42. (1959) Zbl0088.28505MR0110807
  6. Yang C. C., On deficiencies of differential polynomials II, Math. Z. 125 (1972), 107–112. (1972) Zbl0217.38402MR0294642
  7. Yang C. C., Hua X. H., Uniqueness and value-sharing of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 22 (1997), 395–406. (1997) Zbl0890.30019MR1469799
  8. Yang C. C., Yi H. X., Uniqueness Theory of Meromorphic Functions, Science Press & Kluwer Academic Punlishers, Beijing & Dordrecht, 2003. Zbl1070.30011MR2105668
  9. Yi H. X., Uniqueness of meromorphic functions and a question of C. C. Yang, Complex Variables Theory Appl. 14 (1990), 169–176. (1990) Zbl0701.30025MR1048716

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