Which of these statements about biholomorphic functions $f \colon D(0, 1) → D(0, 1)$ is true? 
$f \colon D(0, 1) → D(0, 1)$ is a biholomorphic function.
a) $f$ must be constant
b) $f$ must have a fixed point
c) $f$ must be a rotation
d) $f$ must fix the origin.

Any such map looks like $e^{i\alpha}{(z-a)\over (1-\bar{a}z)}$, $a\in D$, $\alpha\in [0,2\pi]$, so only c is correct option?

It helps to keep in mind the classification of biholomorphic self-maps of $\mathbb D$ (also known as orientation-preserving isometries of the hyperbolic plane in the disk model) into


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*hyperbolic, which look like translation, and have no fixed points in $\mathbb D$ 





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*parabolic, which look like rotation about a boundary point (which is not a point in $\mathbb D$). They have no fixed points either: 





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*elliptic, which look like rotation about a point in $\mathbb D$ (which is a fixed point).



(The author of the materials linked above is Colleen Robles, as far as I can tell.) 
a) clearly false: a bi-(anything) function cannot be constant.
b) c), d) are disproved by an example of  hyperbolic or parabolic isometry
