Let $f(z)=z^n+nwz$ be a complex function with $|w|=1$ and $n>1$ a natural number. Is this function one-to-one inside the unit circle ($|z|<1$)?
ATTEMPT I didn't have a lot of luck checking $f(z_1)=f(z_2)$ and trying to manipulate the equation to get $z_1=z_2$. The factor of $n$ really invites taking the derivative, and I know there's a theorem that states something along the lines of, if $f$ is analytic in $R$ and $f'<>0$ then $f$ is one-to-one in R, but the proof of this theorem that I found relies on complex integration, which we haven't learned yet.
I figure I could try and separate $f$ to real and imaginary components and find the derivative of those, and do some ad-hoc analysis, but is there an easier solution?