Timeline for Do you know important theorems that remain unknown?
Current License: CC BY-SA 4.0
14 events
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| Nov 22, 2021 at 10:33 | comment | added | GH from MO | The condition "bounded" can be omitted, since any continuous function on $\widehat{\mathbb{C}}$ is bounded. For the same reason, the case of $w=\infty$ does not occur, hence it can be disregarded in the proof. Finally, one also needs to assume that $f(\infty)\neq 0$, along with $\infty\not\in E$, in order for the $z_j$'s to lie in $\mathbb{C}$. | |
| Feb 21, 2019 at 20:10 | comment | added | Malik Younsi | @FedorPetrov Thanks! | |
| Feb 21, 2019 at 20:09 | comment | added | Malik Younsi | @OlegEroshkin I was not aware of Shabat's book, thanks for the reference! | |
| Feb 20, 2019 at 21:06 | history | edited | Fedor Petrov | CC BY-SA 4.0 |
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| Feb 20, 2019 at 20:19 | comment | added | Oleg Eroshkin | Shabat's book (Introduction to complex analysis vol.1) has this theorem as a problem. The shortest proof that I know is based on a notion of degree of the map. A map $f\colon S^2\to\mathbb{C}$ is of degree 0. Let $w\in f(S^2)$. If $f$ is holomorphic at $z\in f^{-1}(w)$, then $z$ makes a positive contribution to $\mathrm{deg} f$ since $f$ preserve the orientation. So it must be compensated by contribution of points in $E$. | |
| Feb 20, 2019 at 19:19 | comment | added | Fedor Petrov | @Kalim I modified the proof a bit, hopefully it became better | |
| Feb 20, 2019 at 19:16 | history | edited | Fedor Petrov | CC BY-SA 4.0 |
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| Apr 5, 2018 at 3:01 | comment | added | Piotr Hajlasz | Let us continue this discussion in chat. | |
| Apr 5, 2018 at 2:41 | comment | added | Piotr Hajlasz | I have to think about the example carefully since I want to understand all details. I like your post a lot. There is a result of Salem-Zygmund about holomorphic functions on the unit disc whose boundary values form a Peano type curve. Your example fits into this well. If the example is your original contribution you could expand it and publish in Amer. Math. Monthly. | |
| Apr 5, 2018 at 2:18 | history | edited | Malik Younsi | CC BY-SA 3.0 |
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| Apr 5, 2018 at 0:23 | history | edited | Malik Younsi | CC BY-SA 3.0 |
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| Apr 4, 2018 at 22:43 | history | edited | Piotr Hajlasz | CC BY-SA 3.0 |
Added link to MathSciNet and separated example from the theorem.
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| S Apr 4, 2018 at 21:17 | history | answered | Malik Younsi | CC BY-SA 3.0 | |
| S Apr 4, 2018 at 21:17 | history | made wiki | Post Made Community Wiki by Malik Younsi |