A note on Jordan–von Neumann constant and James constant

Let X be a non-trivial Banach space. L. Maligranda conjectured CNJ(X)⩽1+J2(X)/4 for James constant J(X) and von Neumann–Jordan constant CNJ(X) of X. Satit Saejung gave a proof of it in 2006. In this note, we show that the last step in Satit Saejung's proof is not valid. Using his proof, the result should be . On the other hand, we give a new proof of CNJ(X)⩽1+J2(X)/4. As an application, we give a relation between J(X) and J(lp(X)).