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We prove that the Hilbert functions of codimension four graded GorensteinArtin algebras R/I are unimodal provided I has a minimal generator in degreeless than five. It is an open question whether all Gorenstein h-vectors incodimension four are unimodal. In this paper, we prove that Hilbert functionsof 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 previouslyknown results, we obtain that all Gorenstein h-vectors (1, h_1, h_2,h_3, h_4,....) are unimodal if h_4\leq 3.

On Unimodality of Hilbert Functions of Gorenstein Artin Algebras of Embedding Dimension Four