More on Reverse Triangle Inequality in Inner Product Spaces

Details More on Reverse Triangle Inequality in Inner Product Spaces More on Reverse Triangle Inequality in Inner Product Spaces

2005-06-10

arxiv.org/abs/math/0506198v1

Inter J. Math. Math Sci., 18 (2005), 2883-2893

Mathematics

Functional Analysis

12 pages

Scientific paper

10.1155/IJMMS.2005.2883

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 $(H;< .,.>)$, $r, s>0, p\in(0,s], D=\{x\in H,\|rx-sa\|\leq p\}, x_1, x_2\in D-\{0\}$ and $ \alpha_{r,s}=\min\{\frac{r^2\|x_k\|^2-p^2+s^2}{2rs\|x_k\|}: 1\leq k\leq 2 \}$, then $$\frac{\|x_1\|\|x_2\|-Re< x_1,x_2>}{(\|x_1\|+\|x_2\|)^2}\leq \alpha_{r,s}.$$

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