Letfbe analytic in an unbounded open setDand assume that the cluster setCffat the boundary ∂Dis contained either in a closed setKwhich is convex at infinity or in a compact setK: in both casesC\Kis connected. We prove that if there existsw0∈f(D)\Ksuch thatf-1(w0) contains at leastppoints, then the maximum modulusM(r,f) = sup|z=|f(z)|zεD, must grow at a certain rate at infinity: ifKis unbounded, then; ifKis compact, thenHere c denotes constants and the inequalities hold for large values ofr.