An extension of a simply inequality between von Neumann.Jordan and James constants in Banach spaces

For a non-trivial Banach space X, let J(X), C NJ(X), C (p) NJ (X) respectively stand for the James constant, the von Neumann.Jordan constant and the generalized von Neumann.Jordan constant recently inroduced by Cui et al. In this paper, we discuss the relation between the James and the generalized von Neumann.Jordan constants, and establish an inequality between them: C (p) NJ (X) . J(X) with p . 2, which covers the well-known inequality C NJ(X) . J(X). We also introduce a new constant, from which we establish another inequality that extends a result of Alonso et al.