incidentally, a related (?) theorem I discovered in the late 1980s is this. Let f(z) be a function analytic within in some ball-shaped compact region R of complex z plane with connected interior. THEOREM: If Re(f), Im(f), Re(f), Im(f) have zeros in that order as we walk round the boundary of R, then f() has a zero inside R. Note, am assuming Re(f) has no other zeros, and Im(f) no others, on boundary(R). You may enjoy proving that. At the time I considered it too easy to publish. On the other hand if we keep on finding such "easy" theorems (e.g. the one Andy Latto just proved is now incredibly easy, although it did not look as easy back when I proved it :) ... put them all together in a pile and it gets more interesting. What also might make it more interesting is if we had some sort of strengthened to "if and only if" version of the present post's theorem... -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)