I'm trying to find the radius of the largest disk about the origin so that the map $f(z)=z^2+z$ is injective.
I know $f(0)=0$ and f'(0)=1\neq 0, so there is at least some disk of positive radius where $f(z)$ is injective. Also, $f(0)=f(-1)=0$, so the disk can have radius strictly smaller than $1$.
I say if $f(z)=a$ is injective iff $f(z)-a=0$ has at most one root. The roots are $ -\frac{1}{2}\pm\frac{\sqrt{1+4a}}{2} $ and thus are point on the opposite side of a circle in the complex plane centered at $-1/2$ with radius $|\sqrt{1+4a}/2|$. I'm stuck here. How can I explicitly find the largest radius of the disk around $0$ so that $f$ is injective on that disk? Thank you.