I'm trying to prove that the mapping
$Z=z^m$
for $m \geq \dfrac 12$, transforms the region $r \geq 0$, $0\leq\theta\leq\dfrac{\pi}{m}$ into the upper half plane $Y\geq 0$ where $Z=X+iY.$
My attempt was to write
$z=re^{i\theta}$ so that
$$z^m = r^m\bigg[\cos(\theta m) + i\sin(\theta m)\bigg],$$
and the upper-half region would be controlled by $\operatorname{Im}(z)$, so
$$\operatorname{Im}(z^m) = r^m \sin(\pi)=r^mk$$
so that $k \in [0,1].$
I feel like I am missing why $m$ must be $ \geq \dfrac 12.$
Any help is appreciated.