A simple inequality for the von Neumann–Jordan and James constants of a Banach space

Let C NJ ( X ) and J ( X ) be the von Neumann–Jordan and James constants of a Banach space X, respectively. We shall show that C NJ ( X ) ⩽ J ( X ) , where equality holds if and only if X is not uniformly non-square. This answers affirmatively to the question in a recent paper by Alonso et al. [J. Alonso, P. Martín, P.L. Papini, Wheeling around von Neumann–Jordan constant in Banach spaces, Studia Math. 188 (2008) 135–150]. This inequality looks quite simple and covers all the preceding results. In particular this is much stronger than Maligrandaʹs conjecture: C NJ ( X ) ⩽ J ( X ) 2 4 + 1 .