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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∆ (4)<br />
∆ (1) ∆ (2)<br />
As the line segments in the interior <strong>of</strong> ∆ also occur as their reversed paths, we<br />
have<br />
� 4�<br />
�<br />
f = f,<br />
so that<br />
��<br />
�<br />
�<br />
�<br />
∂∆<br />
∂∆<br />
Choose j ∈ {1,2,3,4} such that � � �<br />
that �� ���<br />
�<br />
�<br />
f�<br />
� ≤<br />
∂∆<br />
j=1<br />
4�<br />
��<br />
�<br />
�<br />
�<br />
j=1<br />
∂∆ (j)<br />
∂∆ (j)<br />
�<br />
�<br />
f�<br />
� .<br />
∆ (3)<br />
29<br />
∂∆ (j) f � � is largest, and set ∆1 := ∆ (j) . It follows<br />
� ��<br />
� �<br />
f�<br />
� ≤ 4�<br />
�<br />
also, note that ℓ(∂∆1) = 1<br />
2 ℓ(∂∆).<br />
Repeat this argument with ∆1 in place <strong>of</strong> ∆, and obtain a triangle ∆2 ⊂ ∆1 with<br />
and �� ���<br />
so that �� ���<br />
∂∆1<br />
�<br />
�<br />
f�<br />
� ;<br />
ℓ(∂∆2) = 1<br />
2 ℓ(∂∆1) = 1<br />
4 ℓ(∂∆)<br />
∂∆<br />
∂∆1<br />
� ��<br />
� �<br />
f�<br />
� ≤ 4�<br />
�<br />
Inductively, we obtain triangles<br />
� ��<br />
� �<br />
f�<br />
� ≤ 4�<br />
�<br />
∂∆1<br />
∂∆2<br />
�<br />
�<br />
f�<br />
� ,<br />
� ��<br />
� �<br />
f�<br />
� ≤ 16�<br />
�<br />
∆ ⊃ ∆1 ⊃ ∆2 ⊃ ···<br />
∂∆2<br />
�<br />
�<br />
f�<br />
� .