  We establish sharp Sobolev inequalities of order four on Euclidean d-balls
for d greater than or equal to four. When d=4, our inequality generalizes the
classical second order Lebedev-Milin inequality on Euclidean 2-balls. Our
method relies on the use of scattering theory on hyperbolic d-balls. As an
application, we charcaterize the extremals of the main term in the
log-determinant formula corresponding to the conformal Laplacian coupled with
the boundary Robin operator on Euclidean 4-balls.
