= Solution
Put $\eta=(\gamma_2-a)/(\gamma_1-a)$. The strict inequality gives $|\eta-1|<1$, so $\operatorname{Re}\eta>0$. This is a closed piecewise-smooth curve in the right half-plane, and part (b) gives $n(\eta,0)=0$. Since $\gamma_2-a=(\gamma_1-a)\eta$, part (a) yields
$$
n(\gamma_2,a)=n(\gamma_1,a)+n(\eta,0),
\qquad \boxed{n(\gamma_2,a)=n(\gamma_1,a).}
$$
This is the <dominated perturbation preserves winding number> principle, here derived by logarithmic differentiation.
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