The Dunkl-Williams inequality with n elements in normed linear spaces

Abstract

Abstract. In this paper we establish a generalization of the Dunkl-Williams inequality for finitely many elements in a normed linear space. As a consequence, we get some recently obtained results on the generalized triangle inequality and its reverse inequality. The case of equality for elements of a strictly convex normed linear space is also considered. (2000): 26D15. Mathematics subject classification

Generalization of the Pecaric-Rajic Inequality in a Quasi-Banach Space

“…Furthermore, the authors [6] also showed that these inequalities imply some refinements of the generalized triangle inequalities obtained by some authors.…”

Refinements of the Dragomir inequality for integrable functions in a normed linear space

“…On the another hand, Pecaric-Rajic [6] obtained the following inequalities which are sharper than inequalities (1.1) above.…”

Dunkl–williams Inequalities for Integrable Functions in Banach Space

“…In [10], the authors considered the case of equality in some generalisations of the Dunkl-Williams inequality for elements of a pre-Hilbert C *module. Recently, Pecaric et al [9] presented the following general Dunkl-Williams inequality for an arbitrary number of finitely many nonzero elements of a normed linear space:…”

“…They also considered conditions for equality in (1.2) and (1.3) to hold in strictly convex normed linear space. In [9], the authors showed that these inequalities imply c 2012 Australian Mathematical Publishing Association Inc. 0004-9727/2012 $16.00 [2] Dunkl-Williams inequalities for integrable functions 299 the following refinements of the generalised triangle inequalities obtained by Kato et al in [6]:…”