Real and Complex Analysis (Rudin)

214 REAL AND COMPLEX ANALYSIS
It is uniform convergence on compact subsets which arises most naturally
in connection with limit operations on holomorphic functions. The term
" almost uniform convergence" is sometimes used for this concept.
10.28 Theorem Suppose fj E H(O), for j = 1, 2, 3, ... , and fj--4 f uniformly on
compact subsets ofO. ThenfE H(O), andfj--4f' uniformly on compact subsets
ofO.
PROOF Since the convergence is uniform on each compact disc in 0, f is
continuous. Let L\ be a triangle in O. Then L\ is compact, so
r f(z) dz :;:: lim r liz) dz = 0,
JM
j -+ 00 Ja."
by Cauchy's theorem. Hence Morera's theorem implies thatf E H(O).
Let K be compact, K c O. There exists an r > 0 such that the union E of
the closed discs D(z; r), for all z E K, is a compact subset of O. Applying
Theorem 10.26 to f - fj, we have
(z E K),
where IlfIIE denotes the supremum of I f Ion E. Sincefj--4 funiformly on E, it
follows thatfj--4 f' uniformly on K.
IIII
Corollary Under the same hypothesis.!)n) --4 fIn) uniformly, as j --4 00, on every
compact set K c 0, and for every positive integer n.
Compare this with the situation on the real line, where sequences of infinitely
differentiable functions can converge uniformly to nowhere differentiable functions!
The Open Mapping Theorem
If 0 is a region andf E H(O), thenf(O) is either a region or a point.
This important property of holomorphic functions will be proved, in more
detailed form, in Theorem 10.32.
10.29 Lemma Iff E H(O) and g is defined in 0 x 0 by
then g is continuous in 0 x O.
. If (Z) - f(w)
g(z, w) = z - w
f'(z)
if w -# z,
if w = z,