Uniqueness and value sharing of meromorphic functions on annuli

Abstract

In this paper, we study meromorphic functions that share only one value on annuli and prove the following results.
1. Let $f(z)$ and $g(z)$ be two non constant meromorphic functions in $\mathbb{A}\left(R_0\right)$, where $1<R_0 \leq+\infty, n \geq 6$. If $f^n f^{\prime} g^n g^{\prime}=1$, then $f \equiv d g$ or $g=c_1 e^{c z}$ and $f=c_2 e^{-c z}$, where $c, c_1$ and $c_2$ are constants and $\left(c_1 c_2\right)^{n+1} c^2=-1$.
2. Let $f(z)$ and $g(z)$ be two non constant entire functions in $\mathbb{A}\left(R_0\right)$, where $1<R_0 \leq+\infty, n \geq 1$. If $f^n f^{\prime} g^n g^{\prime}=1$, then $f \equiv d g$ or $g=c_1 e^{c z}$ and $f=c_2 e^{-c z}$, where $c, c_1$ and $c_2$ are constants and $\left(c_1 c_2\right)^{n+1} c^2=-1$.

Using the results (1) and (2) we prove, let $f(z)$ and $g(z)$ two non constant meromorphic functions on annuli and For $n \geq 11$, if $f^n f^{\prime}$ and $g^n g^{\prime}$ share the same nonzero and finite value $a$ with the same multiplicities on annuli, then $f \equiv d g$ or $g=c_1 e^{c z}$ and $f=c_2 e^{-c z}$, where $d$ is an $(n+1)^{t h}$ root of unity, $c, c_1$ and $c_2$ being constants.