Here is Cauchy's Estimate and its proof from John B. Conway "Functions of One Complex Variable":
And here are Corollary 2.13 and Proposition 1.17:
My questions about the proof of Cauchy's Estimate are:
1- Where has the $i$ that was in Corollary $2.13$ gone?
2- why in the proof 2.14 we are multiplying by $2 \pi r$?
Could someone help me answer these 2 questions please?
For $\Omega$ open connected set s.t. $\overline{D_r(z_0)}\subset\Omega$, $f\in H(\Omega)$ and $n\ge 0$ you have $$|f^{(n)}(z_0)|=\left\vert\frac{n!}{2\pi i}\int_{\vert w-z_0\vert=r}\frac{f(w)}{(w-z_0)^{n+1}}dw \right\vert\le \frac{n!}{2\pi}\frac{\sup_{w\in D_r(z_0)}|f(w)|}{r^{n+1}}2\pi r\\=\frac{n!}{r^n}\sup_{w\in D_r(z_0)}|f(w)|.$$ Remember that $\left|\int_{\gamma}f(z)dz\right|\le\int_{\gamma}|f(z)||dz|$, where $|dz|$ is equal to $ds=\sqrt{x'(t)^2+y'(t)^2}$.