Let $z,w\in\mathbb{C}$ be complex variables and define the function $f:w\mapsto z$ given by \begin{align} z=w+ aw^2+bw\bar{w}+c\bar{w}^2, \end{align} where $a,b,c\in\mathbb{C}$ are complex constants. What is the inverse function $f^{-1}:z\mapsto w\:$? I am not very familiar with complex analysis, so I greatly appreciate any comment or response.
Note 1: We can assume that the function is invertible, so we do not need to worry about the invertibility.
Note 2: I am reading a textbook that claims that the inverse function is \begin{align} w=z - az^2 - bz\bar{z}-c\bar{z}^2 + O(|z|^3), \end{align} and no more information is given.
Hint:
The given function can be decomposed in the form of a system of real equations $$u=P(x,y),\\v=Q(x,y)$$
where $P,Q$ are bivariate quadratic polynomials.
When you vary $u$ and $v$, you obtain pencils of conics. Hence the solutions in $x,y$ are formed by the intersections of two conics, and this leads to a quartic equation, having up to four distinct solutions.
There is a (complicated) analytical solution, and you will have to select branches...