Deciding whether certain sets are or aren't star domains I'm having trouble proving/disproving that the following sets are/aren't star domains:
$$U_1 = \{z \in \mathbb{C}\; | \; |z| < 1, |z+1| > \sqrt{2}\}$$
$$U_2 = \{z \in \mathbb{C}\; | \; \operatorname{Re}(z) < 2, |z-i| > 1\}$$
For $U_2$ it's intuitively clear to me that this is not a star domain. Take any $z_* \in U_2$ and draw 2 tangent lines that are not parallel such that they touch the edge of the circle $B_1(i)$ and cross at $z_*$. Then for any point $z$ "behind" the circle and between those 2 tangents we have $[z_*z] \not\subset U_1$ but I'm having trouble justifying this rigorously.

For $U_2$, your logic is good to describe the entire bad region, but you don't need the entire bad region,  just one point.   So take any $z_*$ in $U_2$,  draw the straight line connecting it to the center of the circle $(0,i)$,  then continue that path for $1.5$ units.  You're guaranteed to still be in the region since you passed the circle of exclusion but could not have reached the line at the real part equaling 2,  and the straight line connecting those two points goes through the circle.
Edit:  I'm not sure about $U_1$.  I originally thought $z=1$ would work, but forgot that's not in the set.
