The Uniform Convexity Of Banach Space And Trigonometric Martingale Inequalities

The history of the studies about geometry of Banach spaces is very long. Among those; studies, Liu and Bek have obtained many important results for the theory and applications of Banach spaces and their geometry on complex number, (see [3],[41])Here, we have investigated the TP modulus of convexity and TP modulus of smoothness, on the one hand, we have defined a class of new spaces called uniformly TP convex ,on the other hand, we have extended martingale inequalities and the martingale spaces.This article is divided into four parts, in the first part, we define the TP modulus of convexity and TP modulus of smoothness of Banach space, and prove that the space which is characterized by uniform convexity is same as the space which is characterized by TP uniform convexity (in isomorphic sense). Then we give TP q-uniformly convex and TP p-uniformly smoothable characterization of the Banach space. At the same time, we prove the famous renormed theorem (in martingale way).In the second part, we give the relationships between some inequalities of trigonometric martingales with values in Banach space and TP uniform convexity of the space. So we characterize the TP q-uniformly convex of the Banach space by inequalities of trigonometric martingales with values in Banach space.In the third part, we consider some trigonometric martingale space and the embedding each other, using these we characterize TP q-uniformly convex of the space.In the fourth part, we study the growth velocity of q-mean square function of trigonometric martingales, using these we characterize TP q-uniformly convex of the space.