ABSTRACT
One of the smaller triangles T1,T2,T3, or T4, hereafter denoted by=1, has the property that ∣∣∣∣∮
/ f (z) dz
∣∣∣∣ ≤ 4 ∣∣∣∣∮ /1 f (z) dz
∣∣∣∣. (8.3) (Why?) Consider the same procedure with =1 as the initial triangle (shaded in Figure 8.6), and of the four smaller triangles coming from=1, let the triangle having property (8.3) be denoted by=2. Then we have∣∣∣∣∮
/ f (z) dz
∣∣∣∣ ≤ 4 ∣∣∣∣∮ /1 f (z) dz
∣∣∣∣ ≤ 42 ∣∣∣∣∮ /2 f (z) dz
∣∣∣∣. Continuing this procedure indefinitely, we generate a sequence of triangles =,=1,=2, . . . such that
∣∣∣∣∮/f (z) dz ∣∣∣∣ ≤ 4n ∣∣∣∣ ∮/nf (z) dz
∣∣∣∣, 2. Pn = (perimeter of=n) = 12n P0,
3. Ln = (length of the longest side of=n) = 12n L0.
