Normal families of meromorphic functions concerning shared functions

Cheng-Xiong Sun

It is mainly proved: Let $\mathfrak{F}$ be a family of meromorphic function in $\mathcal{D}$, $a(z)(\neq0)$ and $b(z)(\not\equiv0)$ be two holomorphic functions on $\mathcal{D}$. Suppose that admits the zeros of multiplicity at least 3 for each function $f \in \mathfrak{F}$. For each $f\in \mathfrak{F}$, if $f=a(z)\Leftrightarrow f'=b(z)$ , then $\mathfrak{F}$ is normal in $\mathcal{D}$. Some example shows that the multiplicity of zeros of $f$ is best in some sense. And the result of paper improve and supplement the result of Lei, Yang and Fang [J. Math. Anal. App. 364 (2010), 143-150].