Can't you just view this as a continuous function on a compactum which implies that it is uniformly continuous? Or do you need some specific rate for the error?

@SeverinSchraven well I originally did try to view it that way (and maybe I'm just getting confused by it all), but I thought that there might be some annoying problems when you'd want to go from a polydisc to an arbitrary compact set?

Maybe it becomes easier if you spell out the precise statement you want to prove, i.e. what is fixed, where the elements live and what type of estimates you want. I would guess that using the continuous function $$ g(\xi, z)= \frac{1}{\prod_{j=1}^n (\xi_j-z_j)} $$ on suitable (compact) domains might do the trick, as you are looking to bound $\vert g(\xi, z)-g(\xi, w) \vert$ and $\vert (\xi,z) - (\xi,w)\vert=\vert z-w\vert$. Thus, uniform continuity of $g$ would solve your problem.

I suspect that you're right @SeverinSchraven, thanks for the advice! So if you'd bear with me, I'd like to show that if $(f_j)$ is a sequence of holomorphic functions on $D$ that is uniformly bounded by $M > 0$, then it is equicontinuous on every compact subset of $D$. This means that for any $K$ compact and $\epsilon > 0$, there is $r > 0$ such that whenever $z, w \in K$ satisfies $|z - w| < r$, then $|f_j(z) - f_j(w)| < \epsilon$ for all $j$. I guess the strategy to do this would be to first take a polydisc $P$ centred at some point $a \in K$ and use the Cauchy integral formula...

... to get the estimate $$|f_j(z) - f_j(w)| \leq \frac{M}{(2\pi)^n} \int_{b_0 P} |g(\xi, z) - g(\xi, w)| |d\xi|$$ for any $z, w \in P$. Would you then choose to take $z, w$ in a smaller polydisc $P'$ so that instead of $g$ being defined on $b_0 P \times P$, it's now defined on $b_0 P \times \overline{P'}$? This would then allow us to use uniform continuity right?

Yes, that looks about right.

I hope that you don't mind moving this to the chat, because Math StackExchange thinks the comments section is getting too long! So then by uniform continuity of $g$ on $b_0 \times P \times P'$, we can find $r > 0$ such that whenever $z, w \in P'$ satisfies $|z - w| < r$, the integral is as small as we want it to be, and so $|f_j(z) - f_j(w)| < \epsilon$ for all $j$.

Then to generalise to the compact set $K$, I guess you'd say that the $P'$ form an open cover of $K$ and so we can extract a finite subcover. Each $P'$ in this finite subcover has a corresponding $r > 0$ so that $(f_j)$ is equicontinuous on each $P'$, but I think I've run into the problem of what happens when the candidate $z, w$ lie in different polydiscs? Even after taking the minimum of all such $r$'s, this doesn't get rid of this problem does it?

How do you do it in a single variable? There should be no difference for this step.

Well the proofs I've seen in one variable have that the choice of $r$ for which $|z - w| < r$ is directly related to the radius of the disc. So for example, for a disc of radius $r$ centred at $a$, you can obtain the estimate $|f(z) - f(w)| \leq 4M|z - w|/r$ for any $z, w$ in the disc of radius $r/2$ centred at $a$. You then take the open cover of discs centred at points in $K$ with radius $r/4$ and then take $\delta$ to be the smaller of of either $r/4$ and $\epsilon r/(4M)$.

Then by the triangle inequality, you can guaratee that any $z, w$ such that $|z - w| < \delta$ will satisfy the estimates we did on each disc and everything else follows.

I guess my problem is that if I argue via uniform continuity of $g$ as above, then I know that if I have $z, w$, in a disc $P'$ satisfying $|z - w| < r$ and everything works out. But wouldn't it be possible for $z, w$ to lie in different polydiscs and not in the intersection of such polydiscs, but yet still satisfy $|z - w| < r$?

Similarly as in one variable, you make the polydiscs smaller than you actually need it from the uniform continuity to prevent that from happening. We are free to choose our polydisc for our cover.

Maybe I am missing something. I am fairly tired :)

I'm also very tired (it's way past midnight where I am!) so I'll try and think about it some more after getting some sleep. Thanks for all your help!

I think I've got it! You were right, it comes down to making things a little smaller a bit earlier on and then also choosing the cover of our compact set to consist of slightly smaller balls as well. This will maintain the uniform continuity estimate while avoiding the problem I mentioned before. Thanks again for all your help!

Well, you did essentially everything yourself :) I'd suggest you spell everything out as an answer to your question. Then the next person can benefit from your work.