  A decade ago, when teaching complex analysis, the third named author posed
the question on whether or not there is an analogue to the Schwarz lemma for
real analytic functions. This led to the note [MT], indicating that it is
possible to have a real analytic automorphism $f$ of $(-1,1)$ with $f'(0)$
arbitrarily large. In this note we provide other families with this property,
and moreover show that we can always find such a function so that $f'(0)$
equals any desired real number. We end with some questions on related problems.
