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Hermite-Hadamard’s inequality and the \(p\)-HH-norm on the Cartesian product of two copies of a normed space. (English) Zbl 1183.26025

Summary: 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.

MSC:

26D15 Inequalities for sums, series and integrals
46B20 Geometry and structure of normed linear spaces
46C50 Generalizations of inner products (semi-inner products, partial inner products, etc.)
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