Coamoebas and line arrangements in dimension two

We show that the number of components of the complement of the closure of a coamoeba of an algebraic curve (f^{-1}(0)) on the complex torus (({mathbb C}^*)^2) is at most two times the area of the Newton polygon of (f) . This is an affirmative answer for the two-dimensional case to a conjecture by Mikael Passare.