I think one can use Sard's theorem to prove the maximum modulus Principle in complex analysis, here is the brief skeleton: 1. let a U be region and f takes on "a" in int(U) as a maximum Modulus point, thus it's locally vanishing to a zero-messurable point, which by sard's theorem, the det f'(a) vanishes too, then if f is a diffeomorphism(analytic functions are diffeomorphisms), then at point a, it will vanish to a 1 or 0-manifold on C(det f'(a) =0, not an onto diffeomorphism that transfers to an 2-manifold), but any conformal mapping has non-vanishable det, (coz it's a 2-manifold on C, for every conformal mapping coz it's angle preserving). thus, conformal maps won't satisfy the maximum Modulus condition; now the only analytic functions left are constant functions, thus only constant functions satisfy the maximum Modulus condition. it's my brief proof sketch. Hope you spot any errors. THANK YOU