Properties of Generalized Modulus of Smoothness and Generalized Modulus of Convexity - ZHAO Liang; YU Yang (Journal of Harbin University of Science and Technology)
Based on the definitions of generalized modulus of smoothness, the relation between generalized modulus of smoothness and t in the Banach spaces is studied, which proves three equivalent conditions of uniform normal structure and four equivalent propositions of generalized modulus of smoothness. In addition, which is proved that the Banach space and the super reflexive Banach space satisfy conditions of limt→0ραX（t）t12andραX（t）<α+32tω（x）－1，tω（X）≤1 have uniform normal structure.ραX（t）andω（X）are generalized 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.