Maybe you can use Rouche's theorem? It seems possible that
If this is true, then you can conclude that and have the same number of roots, which would be 1.
Update: This turns out to be false, for .
You could also try to use the argument principle to measure the number of roots in the disk, but the function might be a little nasty to deal with. Putting ,
gives the number of roots in the disk. If you can evaluate this, then you are done.
Maybe somebody has a cleaner way of doing this.