More on reverse triangle inequality in inner product spaces

A H Ansari,M S Moslehian

doi:10.1155/ijmms.2005.2883

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 , then .

Highlights

It is interesting to know under which conditions the triangle inequality went the other way in a normed space X; in other words, we would like to know if there is a positive constant c with the property that c n k=1 xk
N k=1 xk for any finite set x1, . . . , xn ∈ X
The first authors investigating reverse of the triangle inequality in inner product spaces were Diaz and Metcalf [2] by establishing the following result as an extension of an inequality given by Petrovich [8] for complex numbers

Summary

Introduction

It is interesting to know under which conditions the triangle inequality went the other way in a normed space X; in other words, we would like to know if there is a positive constant c with the property that c n k=1 xk. The first authors investigating reverse of the triangle inequality in inner product spaces were Diaz and Metcalf [2] by establishing the following result as an extension of an inequality given by Petrovich [8] for complex numbers. Inequalities related to the triangle inequality are of special interest (cf [6, Chapter XVII]) They may be applied to get interesting inequalities in complex numbers or to study vector-valued integral inequalities [4, 5]. 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}, x1. The reader is referred to [3, 9] for the terminology on inner product spaces

Reverse of triangle inequality

Reverses of Schwarz inequality