Solution
= Solution
Suppose for contradiction that $f$ has no zero in the closed disc. Then
$$
F(s,t)=f(se^{2\pi it})
$$
is a homotopy through loops in $\mathbb C\setminus\{0\}$. At $s=1$ it is $\alpha$, while at $s=0$ it is the constant loop with value $f(0)$. A constant loop has winding number zero, so homotopy invariance gives
$$
w(\alpha;0)=0,
$$
contrary to the hypothesis. Hence
$$
\boxed{\text{there exists }z\in D\text{ with }f(z)=0}.
$$