Consider the following exercise from the book Complex Analysis Stein and Shakarchi
I already proved that $\varphi(z)=z+a_nz^n+O(z^{n+1})$ and $\varphi_k(z)=z+ka_nz^n+O(z^{n+1})$, but I do not see how that with the Cauchy inequalities proves the linearity of $\varphi$.
The Cauchy inequalities corollary states:
Note that, for each $k\in\Bbb N$, $\sup_{z\in\Omega}|\varphi_k(z)|\leqslant\sup_{z\in\Omega}|z|$. So, if $S=\sup_{z\in\Omega}|z|$, you have $\sup_{z\in\Omega}|\varphi_k(z)|\leqslant S$. But then$$(\forall k\in\Bbb N):\left|\varphi_k^{(n)}(z_0)\right|\leqslant\frac{n!S}{R^n}.$$This is impossible, since $\varphi_k^{(n)}(z_0)=n!ka_n$, and so $\lim_{k\to\infty}\left|\varphi_k^{(n)}(z_0)\right|=\infty$.