It is well known that if a family of meromorphic functions is not normal ( a family of meromorphic functions is said to be normal if each sequence of functions in the family has a subsequence which converges locally uniformly to a limit function which is either meromorphic or identically $\infty$ ), then the corresponding family of derivatives may or may not be normal.
For example, $\mathcal{F}:=\{f_n= nz, n\in\mathbb{N}\}$ is not normal on $|z|<1.$ However, the corresponding family of derivatives $\mathcal{F'}=\{n\}$ is normal on $|z|<1.$
Here is my problem:
I am looking for a family of meromorphic functions whose each zero is of multiplicity $2$ and which is not normal on $|z|<1.$ But the corresponding family of derivatives is normal.
Any help shall be largely appreciated.
The answer to this question is negative. This follows easily from the following result of Chen and Lappan (Adv. Math.,vol. 24, 1996, 517-524):
Let $\mathcal{F}$ be a family of meromorphic functions in a domain $D$ such that each $f\in\mathcal{F}$ has zeros of multiplicity at least $k+1,$ where $k$ is a positive integer. If the family $\mathcal{G}=\left\{f^{(k)}:f\in\mathcal{F}\right\}$ is normal in $D,$ then $\mathcal{F}$ is also normal in $D.$