I am trying to solve the following problem:
Let $U\subset \mathbb{C}$ be simply connected. Show that the boundary of $U$ is regular.
Here, we say a boundary $\partial U$ is regular if every point $z\in\partial U$ has the property:
There is a function $\beta$ defined in $D\cap \overline{U}$ where $D$ is a disk about $z$ such that
- continuous and subharmonic on $D\cap U$ ,
- $\beta(z)=0$,
- $\beta(z)<0$ in $D\cap U-\{z\}$.
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I followed the hint: Assume the boundary point is $0$. For simply connected subset we can define $$L(z)=\int^{z} \frac{1}{w}\, dw$$ Then take the function as the real part of $1/L(z)$ $$\tilde{\beta}(z)=\Re \left(\frac{1}{L(z)}\right)=\frac{\log (|z|)}{(\log(|z|))^2+(\arg z)^2}$$ Take the $$\beta(z)=\limsup_{w\to z} \tilde{\beta}(w)$$ so we have $\beta(0)=0$ and $\beta(z)<0$ for other point $z$.
Why we take the upper limit? Is this right?