Normal Families of Meromorphic Functions Sharing a Holomorphic Function

Cheng-Xiong Sun

Let $k$ be a positive integer, and $m$ be an even number. Suppose that $a(z)(\not\equiv 0)$ is a holomorphic function with zeros of multiplicity $m$ in a domain $D$. Let $\cal F$ be a family of meromorphic functions in a domain $D$ such that each $f\in\cal F$ have zeros of multiplicity at least $k+1+m$ and poles of multiplicity at least $m+1$. It is mainly proved that for each pair $(f,g)\in\cal F$, if $ff^{(k)}$ and $gg^{(k)}$ share $a(z)$ IM, then $\cal F$ is normal in $D$. This result improves Hu and Meng's results published in Journal of Mathematical Analysis and Applications (2009, 2011), and also Jiang and Gao's result in Acta Matematica Scientia (2012).