An inequality between Jordan–von Neumann constant and James constant

Let X be a non-trivial Banach space. L. Maligranda conjectured C N J ( X ) ≤ 1 + J ( X ) 2 / 4 for James constant J ( X ) and von Neumann–Jordan constant C N J ( X ) for X . Recently, J. Alonso et al. gave a proof of it and conjectured that C N J ( X ) ≤ J ( X ) is also valid. In this paper, we show that this conjecture is true.