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International Journal of Mathematics and Mathematical Sciences
Volume 2005, Issue 18, Pages 2883-2893
http://dx.doi.org/10.1155/IJMMS.2005.2883

More on reverse triangle inequality in inner product spaces

Department of Mathematics, Ferdowsi University, P.O. Box 1159, Mashhad 91775, Iran

Received 8 February 2005; Revised 17 May 2005

Copyright © 2005 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Refining some results of Dragomir, several new reverses of the generalized triangle inequality in inner product spaces are given. Among several results, we establish some reverses for the Schwarz inequality. In particular, it is proved that if a is a unit vector in a real or complex inner product space (H;.,.), r,s>0, p(0,s], D={xH,rxsap}, x1,x2D{0}, and αr,s=min{(r2xk2p2+s2)/2rsxk:1k2}, then (x1x2Rex1,x2)/(x1+x2)2αr,s.