I'm trying to see if the function:
$z \mapsto z^n+\exp(ia) \cdot nz$
is an injective function at the open unit circle.
Please help.
I'm trying to see if the function:
$z \mapsto z^n+\exp(ia) \cdot nz$
is an injective function at the open unit circle.
Please help.
$z^n+nze^{ia}=w^n+nwe^{ia}\Longrightarrow (z-w)(z^{n-1}+z^{n-2}w+...+zw^{n-2}+w^{n-1})=-ne^{ia}(z-w)$
If $\,z\neq w\,$ then $\,z^{n-1}+z^{n-2}w+...+zw^{n-2}+w^{n-1}=-ne^{ia}$
But, assuming $\,a\in\Bbb R\,$, we get that the RHS's module is $\,n\,$, whereas the LHS's module is $\,|z^{n-1}+z^{n-2}w+...+zw^{n-2}+w^{n-1}|<1+1+...+1 = n$