For all open Riemann surface N and real number 2 (0, /2), we construct a conformal minimal immersion X = (X1,X2,X3) :
N ! R3 such that X3+tan()|X1| : N ! R is positive and proper. Furthermore, X can be chosen with an arbitrarily prescribed flux map.
Moreover, we produce properly immersed hyperbolic minimal surfaces with non-empty boundary in R3 lying above a negative
sublinear graph.