Hi there,

I have the following problems on my hand:

Given arbitrary positive sub-harmonic function $l(x,y)$ on plane, for all analytic functions: $f=a(x,y)+ib(x,y)$ on complex plane, consider the equation:

$|f|^2=l$, i.e. $a^2+b^2=l$

1) for what kind of $l(x,y)$, the equation has $C^2$ solution?

2) for what kind of $l(x,y)$, the *number of solutions is finite*, after quotient of $S^1$ action: $e^{i\theta}\cdot f$ ?

Thanks!