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It is well known that if a family of meromorphic functions is not normal (a family is said to be normal if each sequence in 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.

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.

It is well known that if a family of meromorphic functions is not normal (in the sense of Montel) 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.

It is well known that if a family of meromorphic functions is not normal (a family is said to be normal if each sequence in 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.

It is well known that if a family of meromorphic functions is not normal (in the sense of Montel) 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.

It is well known that if a family of meromorphic functions is not normal (in the sense of Montel) 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.