I'm trying to get some practice using the Maximum Modulus theorem, and want to use it to conclude that the maximum of $(x^2-y^2-1)^2+4x^2y^2$ occurs at $x=0$, $y=\pm 1$, supposing $x^2+y^2\leq 1$.
My thinking is I want to find some suitalbe complex function $f(z)$ such that $|f(z)|^2=(x^2-y^2-1)^2+4x^2y^2$, and then apply the Maximum Modulus principle, since I know the maximum will occur somewhere on the boundary $x^2+y^2=1$. However, I can't find such a function, so maybe I"m approaching it incorrectly.
I did manage to find that $ z^2-1=(x+iy)^2-1=(x^2-y^2-1)+2xyi $ which looked somewhat close, but no cigar. How can this be done better? Thanks.