CVXI6

Give an upper bound for $\int_C \frac{dz}{z^4}, C\,$ is the line segment from $i$ to 1.

The length of the curve $L\,$ is $\sqrt{2}$ .

$\int_C \frac{dz}{z^4} \le \max_{z\isin C} \left|\frac{1}{z^4}\right| \cdot \mathrm{length}(C)\,$

$=\frac{1}{\min_{z\isin C}\left|z^4\right|} \sqrt{2} = \frac{\sqrt{2}}{\left|(\frac{1}{2}+i\frac{1}{2})^4\right|}\,$

$=2^{5/2}\left|\left[e^{i\frac{\pi}{4}}\right]^{-4}\right| = 2^{5/2}\left|e^{-i\pi}\right| = 2^{5/2}\,$