My aim is to show that the function $$f(x+iy)=x^3+ax^2y+bxy^2+cy^3$$ is analytic **only if** $a=3i$ , $b=-3$ , $c=-i$.

For these values of $a$, $b$ and $c$, that $f$ is analytic follows from Cauchy-Riemann equation.

But I unable to test the **only if** condition.