Meromorphic Functions Sharing a Small Function

Abstract

We will study meromorphic functions that share a small function, and prove the following result: let $f\left(z\right)$ and $g\left(z\right)$ be two transcendental meromorphic functions in the complex plane and let $n\ge 11$ be a positive integer. Assume that $a\left(z\right)\left(\not\equiv 0\right)$ is a common small function with respect to $f\left(z\right)$ and $g\left(z\right)$. If $f^n{f}^{\prime }$ and $g^n{g}^{\prime }$ share $a\left(z\right)$ CM, then either $f^n\left(z\right)f^{\prime }\left(z\right)g^n\left(z\right)g^{\prime }\left(z\right)\equiv a^2\left(z\right)$, or $f\left(z\right)\equiv tg\left(z\right)$ for a constant satisfying $t^{n+1}=1$. As applications, we give several examples.