• Research Article
• Open Access

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

Journal of Inequalities and Applications20092009:137301

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

• Accepted: 18 November 2009
• Published:

## 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

• Linear Combination
• Lower Bound
• Complex Number
• Convex Function
• Linear Space

## 1. Introduction

In the recent paper , 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 :

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

where and .

Notice that, in , 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 :

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.

Let be a normed linear space over the real or complex number field . If and for with , then

Proof.

Observe that, for any fixed , , we have
Taking the norm in (2.2) and utilizing the triangle inequality, we have

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.

Remark 2.2.
1. (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 .

2. (ii)

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

3. (iii)
Further to (ii), if , writing and , we have

which holds for any nonzero vectors The first inequality in (2.6) was obtained by Mercer in .

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

which, in turn, is a refinement of the Dunkl-Williams inequality :

Theorem 2.3.

Let be a normed linear space over the real or complex number field . If and for with , then

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.

Let be a normed linear space over the real or complex number field . If for with , then

Proof.

Letting and by using the second inequality in (2.9), we have
On the other hand, letting and by using the first inequality in (2.9), we have

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  by Kato et al.

Theorem 2.6.

Let be a normed linear space over the real or complex number field . If for with and , then

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 .

## 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

(1)
Department of Information and Mathematics Sciences, College of Science, China Jiliang University, Hangzhou, 310018, China
(2)
Department of Mathematics, Tunghai University, Taichung, 40704, Taiwan
(3)
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China

## References 