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Suppose that $f$ is a Holomorphic function defined on an open set U of the complex plane containing $cl(B(0, 1))$. Let $S^1 = {z ∈ C : |z| = 1} = ∂B(0, 1)$.

Show that if $f(S^1)$ is an ellipse and f restricted on $S^1$ is injective, then $f$ is injective on $cl(B(0, 1))$.

I have a hint to apply the argument principle to $f(z) − w_0$ for $w_0 ∈ C$

I feel as though it may be similar to this thread: A holomorphic function $f$, injective on $\partial D$, must be injective in $\bar{D}$?

However, I'm not sure why we are told extra information such as $f(S^1)$ is an ellipse. Would that change the question? I'm worried that I have completely the wrong idea. Can someone please also provide me a proof of the injective part? I'm struggling to understand and it would be highly appreciated, since I've been researching for the last two hours but with not any success.

Harish Chandra Rajpoot
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yw_2003
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  • your question is a duplicate of your link (though I suggest reading: https://math.stackexchange.com/questions/744812/one-one-analytic-functions-on-unit-disc ) if you are going to use Jordan Curve Theorem. The information that $f(S^1)$ is an ellipse may allow you to bypass JCT. – user8675309 Nov 30 '22 at 20:03
  • @user8675309 Ah I thought so. How would I prove the injective part please? – yw_2003 Nov 30 '22 at 20:06
  • @user8675309 Can I also ask why I can skip the Jordan Curve Theorem part sorry? – yw_2003 Nov 30 '22 at 20:09
  • The primary result you need is that a Jordan Curve has a winding number modulus of $1$ for all points in the bounded component ('interior') of its complement (then apply Argument Principle). The fact $f(S^1)$ is an ellipse tells you a ton of information -- i.e. there are two components to the complement of $f(S^1)$ and this allows direct winding number calculation (e.g. suppose $0$ is in the bounded component and then homotopy to $S^1$ ). Please read my suggested link -- there was less than 4 minutes between my comment and your follow up question so you did not digest it before commenting. – user8675309 Nov 30 '22 at 20:14

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