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

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