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In this article, some necessary and sufficient conditions are shown in order that weighted inequality of the form ∫{f^*>λ}Φ1(λw1)w2dP≤C∫Ω^Φ2(C|f∞|w3)w4dP holds a.e. for uniformly integrable martingales f =(fn)n≥0 with some constant C > 0,where Φ1,Φ2