Why are conformal mappings necessarily 1 to 1?

Question

Say, by the Riemann Mapping Theorem, there exists a biholomorphic, conformal mapping from the upper half plane to the (open) unit disk (since the UHP is simply connected and is not the entire complex plane.)

This mapping, call it $f(z)$, is intuitively an onto-mapping, since we can shrink the points in the UHP by suitable scaling to fit into the unit disk in the $w$-plane.

But why would such a mapping necessarily be one-to-one?

The UHP is so much bigger than the unit disk that...shouldn't we run into the situation were $z_1 \ne z_2$, but $f(z_1) = f(z_2)$?

Thanks,

Edit: I'm wondering whether the condition $f'$ not zero (non-vanishing derivative) makes the mapping one-to-one, but I am doubtful; I think the non-vanishing derivative only makes the mapping locally invertible, by the inverse function theorem. But I guess we can use this argument for all of the UHP, and conclude the mapping from the UHP to the unit disk is invertible, one-to-one, and onto. Am I close?

Answer 1

Isn't bijectivity part of the definition of "biholomorphic"? In the particular situation you described, mapping the upper half-plane to the unit disk, instead of invoking the Riemann mapping theorem, you might as well look at a specific such mapping, say $$ z\mapsto\frac{z-i}{z+i},$$ check by calculation that it's one-to-one, and draw some pictures to see how this "compression" of the half-plane into the disk works.

Answer 2

While it was shown by explicit example that a particular conformal mapping from the upper half plane to the unit disk is one-to-one, the more general question of "why conformal mappings are one-to-one" was not addressed in detail. It was mentioned in a comment that conformal mappings are "locally linear", heuristically implying that they are locally one-to-one. While this may be good intuition to keep in mind, it is not precise. In the edit of the question, it is stated that "I think the non-vanishing derivative only makes the mapping locally invertible". Indeed: consider the function $z \mapsto z^2$ restricted to the annulus $A = \{ z : 1 < |z| < 2 \}$. On $A$, this mapping is conformal, but not one-to-one. Let us prove the following:

If $f$ is conformal in an open set $G$, then for each $a \in G$ there exists an $r > 0$ such that the restriction of $f$ to $D(a;r)$ is one-to-one. [$D(a;r)$ represents the open disk centered at $a$ of radius $r$.]

Take $a \in G$ arbitrarily and find $R > 0$ such that 1) $\bar{D}(a;R) \subset G$ and 2) $f(z) - f(a)$ is never zero in the set $\bar{D}(a;R) \backslash \{ a \}$. We must be allowed point 2) by the identity theorem: if we could not find any such $R$ we would be able to construct a sequence $\{ z_n \}$, converging to $a \in G$, on which $f-f(a)$ vanishes. This would imply $f=f(a)$ in a neighborhood of $a$, in turn implying $f'(a) = 0$, which is not allowed by the conformality of $f$. In the following we'll use the result (Rouché's theorem):

Let $f$ and $g$ be holomorphic inside and on a contour $\gamma$ and suppose that $|f(z)| > |g(z)|$ on $\gamma$. Then $f$ and $f+g$ have the same number of zeroes inside $\gamma$.

With this in mind, define $m = \inf \{ |f(z)-f(a)| : z \in C(a;R) \} > 0$, where $C(a;R)$ represents the circle of radius $R$ around $a$. Rouché's theorem now implies that for any $w \in D(f(a);m)$, the functions $f(z)-f(a)$ and $f(z)-w$ have the same amount of zeroes in the domain $D(a;R)$, that is, precisely one. Finally, by continuity of $f$ at $a$, we can always find an $r > 0$ with $r < R$ such that $f(D(a;r)) \subset D(f(a);m)$. Then it is not hard to see that $f$ restricted to $D(a;r)$ is one-to-one.

NOTE: this proof is an exercise, 16.5, in Priestley's book Introduction to Complex Analysis. All the elements that went into the above proof are heavily inspired by the preceding chapters of the book.