Suppose $f(z)=z^2\frac{z-a}{1-\overline{a}z}$, where $a\in\mathbb{C}$. Prove that $f:\mathbb{S}^1\to\mathbb{S}^1$ is a homeomorphism if and only if $|a|\geq3$.
My idea: Surjectivity is quite evident. But when dealing with injectivity, I have encountered difficulty. In complex analysis, it's well known that if $f'(x_0)\neq 0$, then $f$ is locally conformal at $x_0$. However, we have to deal with the whole $\mathbb{S}^1$ here. I tried to compute directly, but in vain. My teacher gave a hint that argument of principle might be helpful. However, I stil don't know on which area should we use this theorem.
Any solution or hint is highly appreciated!