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.


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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