Geodesic
On unicity of meromorphic functions due to a result of Yang - Hua
Archivum mathematicum, Tome 43 (2007) no. 2, pp. 93-103
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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.
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.
Classification : 30D35
Keywords: entire functions; meromorphic functions; value sharing; unicity 
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     title = {On unicity of meromorphic functions due to a result of {Yang} - {Hua}},
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}
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Bai, Xiao-Tian; Han, Qi. On unicity of meromorphic functions due to a result of Yang - Hua. Archivum mathematicum, Tome 43 (2007) no. 2, pp. 93-103. http://geodesic.mathdoc.fr/item/ARM_2007_43_2_a1/

[1] Clunie J.: On a result of Hayman. J. London Math. Soc. 42 (1967), 389–392. | MR | Zbl

[2] Fang M. L., Qiu H. L.: Meromorphic functions that share fixed-points. J. Math. Anal. Appl. 268 (2000), 426–439. | MR | Zbl

[3] Hayman W. K.: Meromorphic Functions. Clarendon Press, Oxford, 1964. | MR | Zbl

[4] Hayman W. K.: Research Problems in Function Theory. Athlore Press (Univ. of London), 1967. | MR | Zbl

[5] Hayman W. K.: Picard values of meromorphic functions and their derivatives. Ann. of Math. 70 (1959), 9–42. | MR | Zbl

[6] Yang C. C.: On deficiencies of differential polynomials II. Math. Z. 125 (1972), 107–112. | MR | Zbl

[7] Yang C. C., Hua X. H.: Uniqueness and value-sharing of meromorphic functions. Ann. Acad. Sci. Fenn. Math. 22 (1997), 395–406. | MR | Zbl

[8] Yang C. C., Yi H. X.: Uniqueness Theory of Meromorphic Functions. Science Press & Kluwer Academic Punlishers, Beijing & Dordrecht, 2003. | MR | Zbl

[9] Yi H. X.: Uniqueness of meromorphic functions and a question of C. C. Yang. Complex Variables Theory Appl. 14 (1990), 169–176. | MR | Zbl