The Schwarz-Pick lemma,
for f analytic and bounded, |f|<1, in the disk |z|<1, is refined:
where Φ(z, r) is a quantity determined by the non-Euclidean area of the image of
and ψ(z, r) is that determined by the non-Euclidean length of the image of the boundary of D(z, r). The multiplicities in both images by f are not counted.