The Modulus Of Convexity Of Quasi-Banach Space And Special Martingale Inequality

This article is divided into four parts. In the first pa.rt,we clolinc the TC inodnhis of convexity and TC modulus of smoothness of quasi-Baiiach space, and prove that the space which is characterized by uniform convexity is same as the space which is cliaracteri/,ed by uniform TC convcxity.Then we give several characterizations of q-uniformly TC convex quasi-Banach space.At the same time ,we prove the triionned-theorcm.In the second part, we give the relationships between some inequalities of (*) martingales with values in quasi-Banach space and uniformly TC convex quasi-Banach space. So we characterize the q-uniformly TC convex quasi-Baiiach space by inequalities of (*) martingales with values in quasi-Banach space.In the third part,we study the growth velocity of q-mean square function of (*) martingales , using these we characterize q-uniformly TC convex Banach space.In the fourth part, we consider some (*) martingale spaces and the embedding each other , using these we characterize q-uniformly TC convex quasi-Banach space.