Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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Set K := supζ∈∂D|f(ζ)|. For z ∈ S, we then have<br />
�<br />
|I2| ≤<br />
(|f(e<br />
[0,2π]\J<br />
iθ )|+|f(z0)|)P1(e iθ ,z)dθ<br />
= 1<br />
�<br />
2π<br />
≤ K<br />
�<br />
π<br />
≤ K<br />
πC2 �<br />
[0,2π]\J<br />
[0,2π]\J<br />
[0,2π]\J<br />
≤ 2K<br />
C 2 (1−|z|2 )<br />
(|f(e iθ )|+|f(z0)|) 1−|z|2<br />
|eiθ dθ<br />
−z| 2<br />
1−|z| 2<br />
|eiθ dθ<br />
−z| 2<br />
(1−|z| 2 )dθ, because z ∈ S,<br />
Choose δ ∈ (0,δ0) so small that |z0 −z|< δ for z ∈ D implies<br />
1−|z| 2 < C2 ǫ<br />
2K 2 .<br />
For z ∈ D with |z0 − z|< δ, we then have z ∈ S and hence |I2|< ǫ.<br />
On combining<br />
2<br />
these results, we see that |g(z0)−g(z)|< ǫ.<br />
Definition. Let D ⊂ C be open, and let f : D → C be continuous. We say that f<br />
has the mean value property if, for every z0 ∈ D, there exists R > 0 with BR[z0] ⊂ D<br />
such that<br />
for all r ∈ [0,R].<br />
f(z0) = 1<br />
� 2π<br />
f(z0 +re<br />
2π 0<br />
iθ )dθ<br />
Theorem 14.4. Let D ⊂ C be open, and let f : D → C have the mean value<br />
property such that |f| attains a local maximum at z0 ∈ D. Then f is constant on a<br />
neighbourhood <strong>of</strong> z0.<br />
Pro<strong>of</strong>. Choose R > 0 with BR[z0] ⊂ D such that |f(z0)|≥ |f(z)| for all z ∈ BR[z0]<br />
and f(z0) = 1<br />
� 2π<br />
2π 0 f(z0 + reiθ )dθ for all r ∈ [0,R]. If f(z0) = 0 then the result is<br />
trivial. Otherwise, let<br />
h(z) = |f(z0)|<br />
f(z0) f(z)<br />
and set g := Reh−|h(z0)|. Then g has the mean value property and satisfies<br />
for z ∈ BR[z0]. It follows that<br />
g(z) ≤ |h(z)|−|h(z0)|≤ 0<br />
0 = g(z0) =<br />
� 2π<br />
0<br />
g(z0 +re iθ ) dθ<br />
� �� �<br />
≤0<br />
101