  Green's general hyperplane restriction theorem gives a sharp upper bound for
the Hilbert function of a standard graded algebra over and infinite field K
modulo a general linear form. We strengthen Green's result by showing that the
linear forms that do not satisfy such estimate belong to a finite union of
proper linear spaces. As an application we give a method to derive variations
of the Eakin-Sathaye theorem on reductions. In particular, we recover and
extend results by O'Carroll on the Eakin-Sathaye theorem for complete and joint
reductions.
