- Research Article
- Open Access
On Pečarić-Rajić-Dragomir-Type Inequalities in Normed Linear Spaces
© Zhao Changjian et al. 2009
- Received: 27 April 2009
- Accepted: 18 November 2009
- Published: 24 November 2009
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.
- Linear Combination
- Lower Bound
- Complex Number
- Convex Function
- Linear Space
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.
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.
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 .
Furthermore, if for and , with , the inequality reduces further to inequality (1.1) obtained by Pečarić and Rajić in .
which holds for any nonzero vectors
The first inequality in (2.6) was obtained by Mercer in .
which, in turn, is a refinement of the Dunkl-Williams inequality :
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.
This completes the proof.
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.
This follows much in the line as the proofs of Theorem 2.1 and Theorem 2.4, and so it is omitted here.
In case the multi-index reduces to a single index , after suitable modifications, (2.17) reduces to inequality (1.3) obtained by Dragomir in .
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).
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