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

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.

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 :

(1.1)

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

(1.2)

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

(1.3)

where and .

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]:

(1.4)

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.

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

(2.1)

Proof.

Observe that, for any fixed , , we have

(2.2)

Taking the norm in (2.2) and utilizing the triangle inequality, we have

(2.3)

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

Next, by (2.2) we have obviously

(2.4)

On utilizing the continuity property of the norm we also have

(2.5)

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

Remark 2.2.

• (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) Further to (ii), if , writing and , we have

  

  (2.6)

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]:

(2.7)

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

(2.8)

Theorem 2.3.

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

(2.9)

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

(2.10)

Proof.

Letting and by using the second inequality in (2.9), we have

(2.11)

Hence

(2.12)

Then it follows that

(2.13)

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

(2.14)

Hence

(2.15)

from which we get

(2.16)

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.

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

(2.17)

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].