(a) Prove that if $f: \mathbb{D} \to \mathbb{D}$ is analytic and has two distinct fixed points, then $f$ is identity.
(b) Must every holomorphic function $f: \mathbb{D} \to \mathbb{D}$ have a fixed point?
Notation. $\mathbb{D}$ is the open unit disk.
My attempt.
(a) Let $\displaystyle \psi_{z_{1}} = \frac{z_{1} – z}{1 – \bar{z_{1}}z}$ and $z_{1},z_{2}$ the fixed points of $f$, and define $g: \mathbb{D} \to \mathbb{D}$ by $g(z) = (\psi_{z_{1}} \circ f \circ \psi_{z_{1}}^{-1})(z)$. Since $\psi_{z_{1}}$ is holomorphic and an automorphism of $\mathbb{D}$ (such that $\psi_{z_{1}}^{2} = id$), $g$ maps $\mathbb{D}$ into itself. Note that
$$g(0) = (\psi_{z_{1}} \circ f \circ \psi_{z_{1}}^{-1})(0) = (\psi_{z_{1}} \circ f)(\psi_{z_{1}}(0)) = \psi_{z_{1}}(f(z_{1})) = 0.$$
Since $\psi_{z_{1}}$ is bijective, there is $\alpha$ such that $\psi_{z_{1}}(\alpha) = z_{2}$, moreover, $\alpha = \psi_{z_{1}}^{2}(\alpha) = \psi_{z_{1}}(z_{2})$. Then,
$$g(\alpha) = (\psi_{z_{1}} \circ f \circ \psi_{z_{1}}^{-1})(\alpha) = (\psi_{z_{1}} \circ f)(\psi_{z_{1}}(\alpha)) = \psi_{z_{1}}(f(z_{2})) = \alpha.$$
Also, if $\alpha = 0$, $z_{1} = z_{2}$, a contradiction. By Schwarz lemma, $g(z) = cz$ where $c = e^{i\theta}$. But, since $g(\alpha) = \alpha$, $c\alpha = \alpha$ and so, $c=1$.
(b) I know that, with tha same ideia that I use in (a), if the answer is yes, I can always apply the Schwarz lemma in every holomorphic function from $\mathbb{D}$ to $\mathbb{D}$. So, I think the answer is no, but I could not find a counterexample. I'm trying to get an example where Schwarz lemma fails.
Best Answer
Another Counterexample :
$f(z)=\dfrac{z+1}{2}$