More on Reverse Triangle Inequality in Inner Product Spaces

Abstract

Refining some results of S. S. 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 $(H;<.,.>)$, $r,s>0,p\in (0,s],D=\{x\in H,\Vert rx-sa\Vert \le p\},x_1,x_2\in D-\{0\}$ and ${\alpha }_{r,s}=\min \{\frac{r^2\Vert x_k{\Vert }^2-p^2+s^2}{2rs\Vert x_k\Vert }:1\le k\le 2\}$, then $$\frac{\Vert x_1\Vert \Vert x_2\Vert -Re<x_1,x_2>}{(\Vert x_1\Vert +\Vert x_2\Vert {)}^2}\le {\alpha }_{r,s}.$$

12 pages