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Hermite-Hadamard's Inequality and the p-HH-Norm on the Cartesian Product of Two Copies of a Normed Space

Kikianty, Eder and Dragomir, Sever S (2008) Hermite-Hadamard's Inequality and the p-HH-Norm on the Cartesian Product of Two Copies of a Normed Space. Research report collection, 11 (1).

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Abstract

The Cartesian product of two copies of a normed space is naturally equipped with the well-known p-norm. In this paper, another notion of norm is introduced, and will be called the p-HH-norm. This norm is an extension of the generalised logarithmic mean and is connected to the p-norm by the Hermite-Hadamard's inequality. The Cartesian product space (with respect to both norms) is complete, when the (original) normed space is. A proof for the completeness of the p-HH-norm via Ostrowski's inequality is provided. This space is embedded as a subspace of the well-known Lebesgue-Bochner function space (as a closed subspace, when the norm is a Banach norm). Consequently, its geometrical properties are inherited from those of Lebesgue-Bochner space. An explicit expression of the superior (inferior) semi-inner product associated to both norms is considered and used to provide alternative proofs for the smoothness and reflexivity of this space.

Item Type: Article
Uncontrolled Keywords: Hermite-Hadamard inequality, Cartesian product, generalised logarithmic mean, semi-inner product
Subjects: FOR Classification > 0101 Pure Mathematics
Collections > Research Group in Mathematical Inequalities and Applications (RGMIA)
Depositing User: Research Group in Mathematical Inequalities and Applications
Date Deposited: 09 Feb 2012 23:45
Last Modified: 24 May 2013 04:12
URI: http://vuir.vu.edu.au/id/eprint/17610
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