I have some homework problems from Greene and Krantz' Function Theory of One Complex Variable. They come from Chapter 5. I definitely do not want answers, just light prodding in the right direction.
Let $f_j: D(0, 1)\to\mathbb C$ be holomorphic and suppose that each $f_j$ has at least $k$ roots in $D(0,1)$, counting multiplicities. Suppose that $f_j\to f$ uniformly on compact sets. Show by example that it does not follow that $f$ has at least $k$ roots counting multiplicities. In particular, construct examples, for each fixed $k$ and each $\ell$, $0\le\ell\le k$, where $f$ has exactly $\ell$ roots. What simple hypothesis can you add that will guarantee that $f$ does have at least $k$ roots?
I know we require continuity of the $f_j$ on the boundary for the number of zeroes to be the same in the limit, but I'm not clear on why this is. Presumably, that's the purpose of the question. In another question, the goal is to prove this when the disk and its boundary are in the region of holomorticity of the $f_j$.
I'm tempted to think that the problem we run in to is when the zeroes move to the boundary in the limit, but maybe I'm just not familiar enough to construct an actual example of this. I'm also confused by their use of at least $k$ roots. It seems simplest to start with a sequence of functions that all have exactly $k$ roots, but I'm afraid I'm missing something, and that it will only work if the functions have different numbers of zeroes, for some reason.
Basically, I would like some intuition about how things can go wrong when we only have holomorticity on the interior of the disk, and maybe a (small) clue about the form the sequence will take.
Prove: If $f$ is a polynomial on $\mathbb C$, then the zeroes of $f^\prime$ are contained in the closed convex hull of the zeroes of $f$. (Here the closed convex hull of a set $S$ is the intersection of all closed convex sets that contain $S$.) [Hint: If the zeroes of $f$ are contained in a halfplane $V$, then so are the zeroes of $f^\prime$.
I would really like to use the maximum modulus theorem to say that the zeroes of $f^\prime$ occur at maxima (minima) of $f$, and therefore that these only happen on the boundary of some set $U$ (where $f$ is continuous on $\overline U$ and holomorphic on $U$), but I can't see a way to relate this statement about general bounded domains and convex hulls.
I could be looking at these entirely wrong. If I need to clarify my thoughts or say more, please let me know, and again, I definitely don't want more than little hints. Thanks all.