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On PečarićRajićDragomirType Inequalities in Normed Linear Spaces
Journal of Inequalities and Applications volume 2009, Article number: 137301 (2009)
Abstract
We establish some generalizations of the recent PečarićRajićDragomirtype 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 :
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]:
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]:
The main aim of this paper is to establish further generalizations of these PečarićRajićDragomirtype 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).
Next, by (2.2) we have obviously
On utilizing the continuity property of the norm we also have
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 multiindices 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 MasseraSchäffer inequality [7]:
which, in turn, is a refinement of the DunklWilliams inequality [8]:
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
Hence
Then it follows that
On the other hand, letting and by using the first inequality in (2.9), we have
Hence
from which we get
This completes the proof.
Remark 2.5.
In case the multiindices 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
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 multiindex reduces to a single index , after suitable modifications, (2.17) reduces to inequality (1.3) obtained by Dragomir in [3].
References
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Kato M, Saito KS, Tamura T: Sharp triangle inequality and its reverse in Banach spaces. Mathematical Inequalities & Applications 2007,10(2):451–460.
Dragomir SS: Bounds for the normalised Jensen functional. Bulletin of the Australian Mathematical Society 2006,74(3):471–478. 10.1017/S000497270004051X
Dragomir SS: Generalization of the PečarićRajić inequality in normed linear spaces. Mathematical Inequalities & Applications 2009,12(1):53–65.
Mercer PR: The DunklWilliams inequality in an inner product space. Mathematical Inequalities & Applications 2007,10(2):447–450.
Maligranda L: Simple norm inequalities. The American Mathematical Monthly 2006,113(3):256–260. 10.2307/27641893
Massera JL, Schäffer JJ: Linear differential equations and functional analysis. I. Annals of Mathematics 1958, 67: 517–573. 10.2307/1969871
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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).
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Changjian, Z., Chen, CJ. & Cheung, WS. On PečarićRajićDragomirType Inequalities in Normed Linear Spaces. J Inequal Appl 2009, 137301 (2009). https://doi.org/10.1155/2009/137301
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DOI: https://doi.org/10.1155/2009/137301