If $f: U \to V$ is holomorphic and injective , then $f'(z) \neq 0$ for all $z \in U$

For $z \ne z_0$ and arbitrary $w$ we have $$ |a(z-z_0)^k-w| - |G(z)| \ge |a(z-z_0)^k| - |w| - |G(z)| \\ = |a(z-z_0)^k| \left( 1 - \left|\frac{G(z)}{a(z-z_0)^k}\right| \right) - |w| \, . $$ Now choose $\epsilon > 0$ such that $$ \left|\frac{G(z)}{a(z-z_0)^k}\right| < \frac 12 $$ for $0 < |z - z_0| \le \epsilon$. Then $$ |a(z-z_0)^k-w| - |G(z)| \ge \frac 12 |a(z-z_0)^k| - |w| $$ so that $|G(z)| < |a(z-z_0)^k-w|$ if $|w| < \frac 12 |a| \epsilon^k$ and $|z - z_0| = \epsilon$.