Extension of Dunkl--Williams inequality and characterizations of inner product spaces

Abstract

Given $ p, q \in \mathbb {R} $, we generalize the classical Dunkl--Williams inequality for $p$-angular and $q$-angular distances in inner product spaces. We extend the Hile inequality for arbitrary $p$-angular and $q$-angular distances and study some geometric aspects of a generalization of Dunkl--Williams inequality. Introducing power refinements, we show significant power refinements of the generalized Dunkl--Williams inequality under some mild conditions. Among other things, we give new characterizations of inner product spaces with regard to $p$-angular and $q$-angular distances. In particular, we prove that if $ p, q, r \in \mathbb {R} $, $ q \neq 0$ and $ 0\leq p/q\lt 1 $, then $ X $ is an inner product space if and only if for every $ x, y \in X \setminus \{0\} $, $$ \bigl \lVert \lVert x \rVert ^{p-1} x - \lVert y \rVert ^{p-1}y \bigr \rVert \leq \frac {2^{1/r}\bigl \lVert \lVert x \rVert ^{q-1} x - \lVert y \rVert ^{q-1}y \bigr \rVert }{\bigl [\lVert x \rVert ^{r(q-p)} + \lVert y \rVert ^{r(q-p)}\bigr ]^{\frac {1}{r}}}. $$