Determining whether a family of power series is normal

Question

How should I check whether a given family of power series forms a normal family? I am trying to apply Montel's theorem that says that a family of holomorphic functions is normal iff it is uniformly bounded on every compact set but I couldn't verify this boundedness condition.

For example, I saw a problem in a book asking to show that the family of power series $\sum_{n=0}^{\infty}a_{n}z^{n}$ with $|a_{n}|\leq n^{2}$ is normal in the open unit disk, but I'm not sure how to do it. Does the fact that each of these series converges uniformly on every smaller closed disk centered at 0 help? A similar problem is to determine whether the family of power series $\sum_{n=1}^{\infty}a_{n}z^{n}$ with $|a_{n}|\leq n$ is normal in the open unit disk.

Answer 1

If $|a_n|\leq n^2$ for all $n$, then for all $r<1$, and all $z$ with $|z|\leq r$, we have $\left|\sum\limits_{n=0}^\infty a_nz^n\right|\leq\sum\limits_{n=0}^\infty |a_n||z|^n\leq\sum\limits_{n=0}^\infty n^2r^n=M(r)<\infty.$ Note that $M(r)$ is independent of the particular sequence $(a_n)$.

The same method applies if $|a_n|\leq f(n)$ where $f$ is any function on the nonnegative integers such that $\sum\limits_{n=0}^\infty f(n)r^n<\infty$ for all positive $r<1$. In particular, this would hold for any polynomial bound.

Answer 2

For the example you gave above you can apply Montel's theorem:

$$ |a_n| \le n^2$$ implies $$| \sum a_n z^2 |\le \sum n^2 |z|^n$$ On any compact subset of the disc $| z|$ has a supremum strictly less than 1 therefore the sum will converge with a uniform bound.

You should be able to argue other cases similarly, there are other criterion you can check but usually uniform boundedness on compact subsets is easiest to check.