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 $e^{i\theta}$, 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|.$1 For this equation to have solutions, it is not enough that$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}\$,