Differentiation 383Thus lf(z)I does not attain a maximum at any point z E G.Theorem 6.29 Let f be defined on the closure of some open and bounded
set G C C, · and suppose that:1. f E V(G).
- JJ(z) =f. 0 everywhere in G.
3. f is continuous on 8G.
Then lf(z)I attains its maximum on 8G.Proof The assumptions imply that f is continuous on the compact set G.
Hence lf(z)I is a continuous real-valued function on G. By a well-known
theorem lf(z)I attains its maximum at some point of G. By Theorem 6.28
this maximum value cannot be attained in G. Therefore, max lf(z)I will
be assumed at at least one point of 8G.
This important property is called the principle of the maximum modulus.
Remark In particular, the two preceding theorems are valid for analytic
or conjugate analytic functions in G provided that f z =f. 0 or fz =f. 0 ev-
erywhere in G, respectively. However, for a more general formulation in
those cases, see Section 8.11.
Theorem 6.30 Let f be defined in some open set G CC, and suppose
that:
1. f E V(G).
- JJ(z) =f. 0 everywhere in G.
- f(z.) =f. 0 at each z E G.
Then lf(z)I does not attain a minimum value anywhere in G.
Proof With the same notation as in the proof of Theorem 6.28, suppose
in equation (6.20-2) that A> B, and choose 'I/; such that o: +'I/; = 0 + 7r.
Then we have
f(z + .6.z) = (M -Ar)eiO -Brei(fJ+cx-O) + r(T/ 1 ei.P + 'f/ 2 e-il/J)
Hence for r < M /A we obtain
lf(z + .6.z)I ~ M -Ar+ Br+ r(l"l1I + IT/21)
= M - r(A-B - l"l1l - l"l2DChoosing r even smaller (if necessary) so that A - B - IT/11-IT/21 > O,
it follows that
lf(z + .6.z)I < M = lf(z)I
If A < B, we let (3 - 'I/; = e + 7r, or 'I/; = (3 - e -7r and proceed in the
same manner.