Abstract: We study the relationships between some inequalities of ananlytic martingale with values in complex quasi-Banach space and uniformly PL-convexitiable of the space. We prove weak type (1,1), strong type (p, p) and a Φ inequalities of q-mean sequare funiction of analytic martingales with values in quasi-Banach space. Appling them to give some equivalent conditons of q -uniformly PL-convexitiable of the spaces. At the same time, appling growth velocity of q-mean square funicton of analytic martingales give characterization of q-uniformly PL-convexitiable spaces.

Key words: Complex Quasi-Banach space, q-uniformly PL-convex, analytic martingale