Uniqueness of meromorphic functions concerning differential polynomials (Q2907920)

This paper considers the uniqueness of meromorphic functions concerning differential polynomials and presents a different and very simply method to handle the uniqueness problems concerning three small entire functions. For example, the authors obtain the following theorem:NEWLINENEWLINE Let \(f\) and \(g\) be two non-constant meromorphic functions and let \(\alpha_{j}\not\equiv 0, \infty\), \(j=1,2,3\), be three non-zero distinct entire small functions. Let \(k\) be a positive integer or \(\infty\) satisfying NEWLINE\[NEWLINE \overline{E}_{k)}\left(\alpha_{j} , f^{n}\left(f^{p}-1\right)f^{'}\right)=\overline{E}_{k)}\left(\alpha_{j} , g^{n}\left(g^{p}-1\right)g^{'}\right),~~j=1,2,3, NEWLINE\]NEWLINE where \(n\) and \(p\) are positive integers. Then either \(f\equiv g\) or NEWLINE\[NEWLINE g=\left[\frac{(n+p+1)\left(h^{n+1}-1\right)}{(n+1)\left(h^{n+p+1}-1\right)}\right]^{1/p},~~ f=\left[\frac{(n+p+1)h^{p}\left(h^{n+1}-1\right)}{(n+1)\left(h^{n+p+1}-1\right)}\right]^{1/p}, NEWLINE\]NEWLINE where \(h\) is a non-constant meromorphic function.