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

More on reverse triangle inequality in inner product spaces

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={x∈H,‖rx−sa‖≤p}, x1,x2∈D−{0}, and αr,s=min{(r2‖xk‖2−p2+s2)/2rs‖xk‖:1≤k≤2}, then (‖x1‖‖x2‖−Re〈x1,x2〉)/(‖x1‖+‖x2‖)2≤αr,s.