  We prove that the Hilbert functions of codimension four graded Gorenstein
Artin algebras R/I are unimodal provided I has a minimal generator in degree
less than five. It is an open question whether all Gorenstein h-vectors in
codimension four are unimodal. In this paper, we prove that Hilbert functions
of all artinian codimension four Gorenstein algebras starting with (1,4,10, 20,
h_4,...), where h_4\leq 34 are unimodal. Combining this with the previously
known results, we obtain that all Gorenstein h-vectors (1, h_1, h_2,h_3, h_4,
....) are unimodal if h_4\leq 3.
