Abstract:Based on the definitions of generalized modulus ofsmoothness

the relation between generalized modulus of smoothness and t in theBanach spaces is studied

which proves three equivalent conditions of uniformnormal structure and four equivalent propositions of generalized modulus ofsmoothness. In addition

which is proved that the Banach space and the superreflexive Banach space satisfy conditions of limt→0ραX（t）t12andραX（t）α+32tω（x）－1，tω（X）≤1 haveuniform normal structure.ραX（t）andω（X）aregeneralized modulus of smoothness and weak orthogonal coefficient respectively.Finally

which gives an inequality about the generalized convex modulus when ‖x‖2+‖y‖2=2

x

y∈X.