  We provide generalisations of two Schwarz-type lemmas -- the first a result
of Globevnik and the other due to Ransford and White -- for holomorphic
mappings into the spectral unit ball. The first concerns mappings of the unit
disc in C into the spectral unit ball, while the second concerns self-
mappings. The aforementioned results apply to holomorphic mappings that map the
origin to the origin. We extend these results to general holomorphic mappings
into the spectral unit ball. We also show that our results are sharp.
