Using the estimation lemma (ML inequality) to prove the following inequality.

Question

show that |∮1/z dz| ≤ (3/4) π where γ :[0, 3/8] and γ(t) = e^(−i2πt)?

I know that the ML inequality shows that:

∮f(z)dz ≤ ML(γ) but am struggling to calculate ML

Answer 1

$\gamma$ is the part of the unit circle from $1$ to $-i$ and $|z|=1$ on the circle. So $M=1$. To find $L$ use the formula $L=\int_a^{b}|\gamma '(t)|\, dt$. So we get $L=\int _0^{3/8} |\frac d {dt} e^{-2\pi i t}| \, dt=\int _0^{3/8}2\pi \, dt=3\pi /4$.