Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 2H (Topics in Analysis)

Normalize the loop byChoose a continuous lift $\vartheta :[0,1]\to \mathbb{R}$ satisfyingBecause $\gamma (0)=\gamma (1)$, the difference $\vartheta (1)-\vartheta (0)$ is an integer independent of the chosen lift. The winding number of a continuous closed path is

If $\eta$ is a lift of $\delta /|\delta |$, then $\vartheta +\eta$ is a lift ofTherefore

DefineThe strict inequality $|s\sigma (t)|<|\gamma (t)|$ ensures that $H(s,t)\ne 0$ for every $(s,t)$, while $H(s,0)=H(s,1)$. Thus $H$ is a homotopy through closed loops in $\mathbb{C}\setminus \{0\}$ from $\gamma$ to $\gamma +\sigma$. By homotopy invariance of winding number,

Suppose for contradiction that $f$ has no zero in the closed disc. Thenis 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 givescontrary to the hypothesis. Hence