  Classically, theorems of Fatou and Julia describe the boundary regularity of
functions in one complex variable. The former says that a complex analytic
function on the disk has non-tangential boundary values almost everywhere, and
the latter describes when a function takes an extreme value at a boundary point
and is differentiable there non-tangentially. We describe a class of
intermediate theorems in terms of averaged Julia-Fatou quotients. Boundary
regularity is related to integrability of certain quantities against a special
measure, the so-called Nevanlinna measure. Applications are given to spectral
theory.
