The theme in this article is the inequalities of B_valued martingales and the properties of Banach spaces. In utilising the techniques of rearranged functions, we proved that for a martingale f we have ‖M αf‖ Φ≤C pΦ ‖f # α‖ Φ and the same result for (Mf,S (p) (f)+D ∞), (Mf∧Sf,Mf∧Sf+D ∞). This contains the results in R.L.Long[1], furthermore, some of the inequalities can be used to describe the properties of Banach spaces.
