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In this article, some necessary and sufficient conditions are shown in order that the inequality of the form Φ1(λ)Pu(f*>λ) ≤ Ev(Φ(C|f∞|)) holds with some constant C > 0 independent of martingale f=(fn)n≥0 and λ > 0, where Φ1 and Φ2 are a pair of Young functions, …… a.e.