On an inequality of Bynum and Drew (Q753081)

For the abstract \(L_ p\) space X with \(1<p\leq 2\) it is proved that for all x,y\(\in X\) and \(0<t<1\) \[ \| (1-t)x+ty\|^ 2\leq (1-t)\| x\|^ 2+t\| y\|^ 2-(p-1)t(1-t)\| x-y\|^ 2. \] This result is applied to show the strong uniqueness of best approximations and existence of fixed points for uniformly Lipschitzian mappings and to prove that if \(J_ s(X)\) is the self-Jung constant [see \textit{D. Amir}, Pac. J. Math. 118, 1-15 (1985; Zbl 0529.46011)] then \(J_ s(X)\leq 2/\sqrt{p}\).