An application of Zbagǎnu constant in fixed point theory (Q2248816)

The Zbăganu constant of a Banach space \(X\) \[ C_Z(X) = \sup\left\{\frac{\| x+y\|\,\|x-y\|}{\|x\|^2 + \|y\|^2}\,:\,x, y \in X\setminus \{0\} \right\} \] was introduced in [\textit{G. Zbăganu}, Rev. Roum. Math. Pures Appl. 47, No. 2, 253--258 (2002; Zbl 1052.46009); addendum ibid. 47, No. 4, 523 (2002)] in order to characterize inner product spaces. In the article under review, the author presents an inequality relating the Zbăganu constant and a coefficient of weak orthogonality, \(\mu(X)\), that was introduced in [\textit{A. Jiménez-Melado} et al., Proc. Am. Math. Soc. 134, No. 2, 355--364 (2006; Zbl 1102.46009)]. As a corollary of the inequality, the author proves that, if \(C_Z(X) < 1 + 1/\mu(X)^2\), then \(X\) has normal structure. The author also proves that \(X\) satisfies a condition stronger than normal structure if the Zbăganu constant of \(X\) is less than Bynum's weakly convergent sequence coefficient. \noindent Reviewer's remark: For another proof of the first-mentioned corollary and for some related results, the interested reader may want to look at an article of \textit{E. Llorens-Fuster} [Fixed Point Theory 9, No. 1, 159--172 (2008; Zbl 1158.46009)].