On a generalized Jordan-von Neumann type constant and normal structure

In this paper, we introduce a new geometric constant C (p) - (a,X ) , which is closely related to the generalized Jordan-von Neumann type constant.We show that 2 and (a+2) p 2 p-2 (2 p +a p ) are the upper and lower bound for C (p)- (a,X ) , respectively.Moreover, we obtain that- a, X , where X is the ultrapower space of X .Subsequently, we give some sufficient conditions for normal structure of a Banach space with different constants, such as the generalized James constant, Domínguez-Benavides coefficient and the coefficient of weak orthogonality.