The Schwarz-Pick lemma,
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0004972700002173/resource/name/S0004972700002173_eqnU1.gif?pub-status=live)
for f analytic and bounded, |f|<1, in the disk |z|<1, is refined:
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0004972700002173/resource/name/S0004972700002173_eqnU2.gif?pub-status=live)
where Φ(z, r) is a quantity determined by the non-Euclidean area of the image of
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0004972700002173/resource/name/S0004972700002173_eqnU3.gif?pub-status=live)
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.