The smoothness and the generalized average uniform convexity of Banach spaces

The concepts of generalized average uniform convexity, generalized average local uniform convexity and generalized average weak local uniform convexity are given. The relations among these uniform convexity and uniform smoothness, very smoothness, uniform extreme smoothness and extreme smoothness are studed. If X is uniform smooth, then X~* is generalized average uniform convex; If X~* is generalized average weak local uniform convex, then X is smooth; If X~* is generalized average uniform convex, then X is very smooth and extremely smooth; If X is uniform extremely smooth and X~* is generalized average weak local uniform convex, then X~* is local uniform convex.