Is there any non-normal family $\mathcal{F}$ of meromorphic functions on $|z|<1$ whose each zero has multiplicity $2$ but $\mathcal{F'}$ is normal

Question

It is well known that if a family of meromorphic functions is not normal (a family 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 $\infty$) on some domain, then the corresponding family of derivatives may or may not be normal on that domain.

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 $|z|<1.$ Furthermore, the family $\mathcal{G}:=\{nz^2\}$ and its derivative $\mathcal{G'}=\{2nz\}$ are not normal on $|z|<1.$

Observe that the family $\mathcal{G}$ has a zero of order $2$ at $z=0$ on $|z|<1$ and its corresponding family of derivatives is not normal.

With the above observation in mind, I am curious to know the following:

Does there exist 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.

Answer 1

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 $D.$