 Solution

The continuous closed curve $q(t)={\gamma }_2(t)/{\gamma }_1(t)$ satisfies $|q(t)-1|<1$. It lies in the right half-plane, which admits a single continuous complex argument in $(-\pi /2,\pi /2)$. Since $q$ closes, this argument has equal endpoint values, giving $w(q)=0$. Additivity under multiplication then yieldsThe strict inequality ensures that the quotient stays in a region avoiding zero and admitting a single argument branch.