Let $\displaystyle c,a \in \mathbb{C}$ be constants, let $\displaystyle n,m \in \mathbb{N}$, and let $\displaystyle z$ be a complex variable.

Is it true that:

$\displaystyle \bigl(c(|z|+1)^n +|a| \bigr) \ |z|^m = c_0 (|z|+1)^{n-m} $

For a suitable constant $\displaystyle c_0$?

It would be very helpful if this were true for a proof I'm trying to figure out...

Thanks