More on reverse triangle inequality in inner product spaces - A. H. Ansari; M. S. Moslehian (International Journal of Mathematics and Mathematical Sciences)
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.