  We show that the orthogonal speed of semigroups of holomorphic self-maps of
the unit disc is asymptotically monotone in most cases. Such a theorem allows
to generalize previous results of D. Betsakos and D. Betsakos, M. D. Contreras
and S. D\'iaz-Madrigal and to obtain new estimates for the rate of convergence
of orbits of semigroups.
