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

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*
Xiao-Tian Bai, Qi Han
*

** Address.**

School of Mathematics and System Sciences, Shandong University, Jinan 250100, Shandong, People's
Republic of China

** E-mail.**

xtbai@163.com

k.l.han@tom.com

**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 $a\in\mathbb{C}\backslash\{0\}$ 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 $n\geq
11$ (resp.\ $n\geq 6$). It improves a result of C.\ C.\ Yang and X.\
H.\ Hua. Also, some other related problems are discussed.
.

**AMSclassification. **

Primary 30D35.

**Keywords. **

Entire functions, meromorphic
functions, value sharing, unicity.