  Opial's inequality and its ramifications play an important role in the theory
of differential and difference equations. A sharp unifying generalization of
Opial's inequality is presented that contains both its continuous and discrete
version. This generalization based on distribution functions is extended to the
case of derivatives of arbitrary order. This extension optimizes and improves
the constant as given in the literature. The special case of derivatives of
second order is studied in more detail. Two closely related Opial inequalities
with a weight function are presented as well. The associated Wirtinger
inequality is studied briefly.
