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Take, for example, the simple linear fractional transformation that sends the upper half plane to the unit disk, and the real line to the unit circle.

We know the fact that the upper half plane (UHP) maps to the inside of the circle (and the LHP maps to the outside) is the result of the mapping being continuous -- and that continuous mappings map connected sets to connected sets.

However, is it also an orientation-preserving argument? I.e., if the mapping is holomorphic, then can I say that the UHP maps to the interior of the unit disk because since the UHP was to the "left" of the real line that is traversed from left to right will again be to the left of the image of the real line. And so the image must be the interior of the disk.

Thanks,

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Yes, it is an orientation preserving argument. In fact, there is another map which maps the real line to the circle, but maps the upper half plane to the outside of the unit disc, and the lower half plane to the interior of the unit disc:

$$f(z) = \frac{\bar{z} - i}{\bar{z} + i}.$$

Of course, this is just the Cayley map precomposed with complex conjugation. This map is orientation-reversing (it is anti-holomorphic), which is why the images of the upper and lower half planes swap.

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