A note on Jordan–von Neumann constant and James constant

Let X be a non-trivial Banach space. L. Maligranda conjectured C NJ ( X ) ⩽ 1 + J ( X ) 2 / 4 for James constant J ( X ) and von Neumann–Jordan constant C NJ ( X ) of X. Satit Saejung gave a proof of it in 2006. In this note, we show that the last step in Satit Saejungʹs proof is not valid. Using his proof, the result should be C NJ ( X ) ⩽ 1 + J ( X ) 2 / 4 + J ( X ) 4 + ( 2 − J ( X ) ) 2 − 2 4 . On the other hand, we give a new proof of C NJ ( X ) ⩽ 1 + J ( X ) 2 / 4 . As an application, we give a relation between J ( X ) and J ( l p ( X ) ) .