Abstract: The p-Amemiya norm inequalities for a generalized martingale transform operators are proved.As an application,a new method to prove the p-Amemiya norm inequalities between the maximal functions and the p-mean functions of B-valued martingales is presented.The results depict the p uniform convexity and q uniform smoothness of the underlying Banach spaces.

Key words: martingales transform, p-Amemiya norm, p uniform convexity, q uniform smoothness