For $0<\alpha<1$, we define $\Omega_\alpha$ to be the union of the disc $D(0;\alpha)$ and the line segments from $z=1$ to points of $D(0;\alpha)$. Now, let $r=|z|$. Given $0<\alpha<1$, $\frac{|z-r|}{1-r}$ is bounded in $\Omega_\alpha$.
I thought this problem is very easy, but I cannot prove it.
Something I tried:
Let $z=x+iy$. Thus, $|z-r|=|x-\sqrt{x^2+y^2}+yi|=2[x^2+y^2-x\sqrt{x^2+y^2}]$ Thus,$\frac{|z-r|}{1-r}=2[\frac{r^2-x}{1-r}+x]=x-(r+1)+\frac{1-x}{1-r}$
I am very confused. Any help will be appreciated.