Let B be the set of holomorphic maps f of the unit disk into itself. Thinking of the range disk as having the hyperbolic metric leads to consideration of the functions \(\sigma\) (f)(z), the hyperbolic distance of f(z) to the origin, and \(f^{\#}\) hyperbolic derivative of f given by \(f^{\#}=| f'| /(1-| f|^ 2).\) The author proves that a function in B which extends continuously to the boundary of the disk is absolutely continuous in the hyperbolic sense if and only if the hyperbolic derivative has a harmonic majorant. He also proves that if f is a function in B such that \(f^{\#p}\) has a harmonic majorant for some p, \(0