bound of an analytic function, and a bit on uniform convergence

Another practice exam exercise:

Quote:

Suppose

is an analytic function in

for which

is finite.

**(a)** Prove for

that

.

**(b)** If

is a sequence of analytic functions in

for which

for

, and

as

, prove

converges to

uniformly on compact subsets of

.

I can't do part (a) at all, and I'm not much further along on part (b). If I use the result of (a), all I need is to show that , and then part (b) follows from Weierstrass. But I cannot show that , nor can I show part (a) at all.

I also notice that is not explicitly defined. I presume it's just some arbitrary (but fixed) subset of . But perhaps it's a misprint...? I don't know...

As usual, any help would be appreciated!