Given two rational maps φ and ψ on \PP^1 of degree at least
two, we study a symmetric, nonnegative-real-valued pairing
which is closely related to the canonical height functions $h_\varphi$ and
$h_\psi$ associated to these maps. Our main results show a strong connection
between the value of and the canonical heights of points which
are small with respect to at least one of the two maps φ and ψ.
Several necessary and sufficient conditions are given for the vanishing of
. We give an explicit upper bound on the difference between the
canonical height $h_\psi$ and the standard height $h_\st$ in terms of
, where σ(x)=x2 denotes the squaring map. The pairing
is computed or approximated for several families of rational
maps ψ