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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!

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1 Answer

For this equation to have solutions, it is not enough that $l$ is subharmonic. It must be logarithmically subharmonic which means that $\log l$ is subharmonic. Now if $l$ is logarithmically subharmonic, the nesessary and sufficient condition that $f$ exists in a simply connected domain is that the measure $(2\pi)^{-1}\Delta\log l$ is discrete and integer-valued.

If the solution exists it is always unique up to a constant factor $e^{i\theta}$.

EDIT. To prove sufficiency, one uses the Weierstrass representation. Suppose that $l$ is logarithmically subharmonic and $(2\pi)^{-1}\Delta\log l$ is a discrete integer-valued measure with atoms $m_k$ at the points $a_k$. Consider the Weierstrass product $f$ with zeros $a_k$ of multiplicity $m_k$. This $f$ is an entire function. Now $u:=\log l-\log|f|$ is harmonic, by construction. Let $v$ be such harmonic function that $g:=u+iv$ is entire. Then $l=|f\exp g|.$

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Thank you so much! In this case, we have $log l$ is indeed harmonic. But, how to prove it is sufficient condition? I still have no clue of this.. Do you have any reference? Thanks! –  Jay Dec 13 '12 at 15:47
    
If $f$ is analytic and nonzero, then $\log |f|$ is harmonic, right? Even with zeros, there is Jensen's formula ... en.wikipedia.org/wiki/Jensen%27s_formula –  Gerald Edgar Dec 25 '12 at 20:44
    
thanks a lot! the relation between harmonic function and holomorphic function is essentially important here... –  Jay Jan 25 '13 at 23:26
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