# On Pečarić-Rajić-Dragomir-Type Inequalities in Normed Linear Spaces

- Zhao Changjian
^{1}, - Chur-Jen Chen
^{2}and - Wing-Sum Cheung
^{3}Email author

**2009**:137301

https://doi.org/10.1155/2009/137301

© Zhao Changjian et al. 2009

**Received: **27 April 2009

**Accepted: **18 November 2009

**Published: **24 November 2009

## Abstract

We establish some generalizations of the recent Pečarić-Rajić-Dragomir-type inequalities by providing upper and lower bounds for the norm of a linear combination of elements in a normed linear space. Our results provide new estimates on inequalities of this type.

## Keywords

## 1. Introduction

In the recent paper [1], Pečarić and Rajić proved the following inequality for nonzero vectors , in the real or complex normed linear space :

and showed that this inequality implies the following refinement of the generalised triangle inequality obtained by Kato et al. in [2]:

The inequality (1.2) can also be obtained as a particular case of Dragomir's result established in [3]:

Notice that, in [3], a more general inequality for convex functions has been obtained as well.

Recently, the following inequality which is more general than (1.1) was given by Dragomir [4]:

The main aim of this paper is to establish further generalizations of these Pečarić-Rajić-Dragomir-type inequalities (1.1), (1.2), (1.3), and (1.4) by providing upper and lower bounds for the norm of a linear combination of elements in the normed linear space. Our results provide new estimates on such type of inequalities.

## 2. Main Results

Theorem 2.1.

Proof.

which, on taking the minimum over , produces the second inequality in (2.1).

which, on taking the maximum over , , produces the first part of (2.1) and the theorem is completely proved.

- (i)
In case the multi-indices and reduce to single indices and , respectively, after suitable modifications, (2.1) reduces to inequality (1.4) obtained by Dragomir in [4].

- (ii)
Furthermore, if for and , with , the inequality reduces further to inequality (1.1) obtained by Pečarić and Rajić in [1].

- (iii)

which holds for any nonzero vectors

The first inequality in (2.6) was obtained by Mercer in [5].

The second inequality in (2.6) has been obtained by Maligranda in [6]. It provides a refinement of the *Massera-Schäffer inequality* [7]:

which, in turn, is a refinement of the *Dunkl-Williams inequality* [8]:

Theorem 2.3.

This follows immediately from Theorem 2.1 by requiring for , and letting for ; .

A somewhat surprising consequence of Theorem 2.3 is the following version.

Theorem 2.4.

Proof.

This completes the proof.

Remark 2.5.

In case the multi-indices and reduce to single indices and , respectively, after suitable modifications, (2.10) reduces to inequality (1.2) obtained in [2] by Kato et al.

Theorem 2.6.

This follows much in the line as the proofs of Theorem 2.1 and Theorem 2.4, and so it is omitted here.

Remark 2.7.

In case the multi-index reduces to a single index , after suitable modifications, (2.17) reduces to inequality (1.3) obtained by Dragomir in [3].

## Declarations

### Acknowledgments

The first author's work is supported by the National Natural Sciences Foundation of China (10971205). The third author's work is partially supported by the Research Grants Council of the Hong Kong SAR, China (Project no. HKU7016/07P).

## Authors’ Affiliations

## References

- Pečarić J, Rajić R:
**The Dunkl-Williams inequality with****elements in normed linear spaces.***Mathematical Inequalities & Applications*2007,**10**(2):461–470.MathSciNetMATHGoogle Scholar - Kato M, Saito K-S, Tamura T:
**Sharp triangle inequality and its reverse in Banach spaces.***Mathematical Inequalities & Applications*2007,**10**(2):451–460.MathSciNetView ArticleMATHGoogle Scholar - Dragomir SS:
**Bounds for the normalised Jensen functional.***Bulletin of the Australian Mathematical Society*2006,**74**(3):471–478. 10.1017/S000497270004051XMathSciNetView ArticleMATHGoogle Scholar - Dragomir SS:
**Generalization of the Pečarić-Rajić inequality in normed linear spaces.***Mathematical Inequalities & Applications*2009,**12**(1):53–65.MathSciNetView ArticleMATHGoogle Scholar - Mercer PR:
**The Dunkl-Williams inequality in an inner product space.***Mathematical Inequalities & Applications*2007,**10**(2):447–450.MathSciNetView ArticleMATHGoogle Scholar - Maligranda L:
**Simple norm inequalities.***The American Mathematical Monthly*2006,**113**(3):256–260. 10.2307/27641893MathSciNetView ArticleMATHGoogle Scholar - Massera JL, Schäffer JJ:
**Linear differential equations and functional analysis. I.***Annals of Mathematics*1958,**67:**517–573. 10.2307/1969871MathSciNetView ArticleMATHGoogle Scholar - Dunkl CF, Williams KS:
**A simple norm inequality.***The American Mathematical Monthly*1964,**71**(1):53–54. 10.2307/2311304MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.