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arXiv · 2507.00009

A note on D-functions and P-covariances on Hilbert spaces and related inequalities

Sergio Scarlatti
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Abstract

In this note we first review the concept of D-function, closely connected with Cauchy-Schwarz inequality, and then introduce the notion of P-covariance on a Hilbert space, where $P$ is an orthogonal projection. We show that when P is specialized to be one-dimensional many well-known inequalities such as Buzano, Richard and Walker inequalities are simple consequences of P-covariance inequalities and their relation with D-functions. By means of these concepts enhancements of the previous mentioned inequalities are also established with minimum effort. A more thorough analysis of Walker inequality is presented jointly with some novel financial considerations as well as a new refinement of Holder inequality.

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Sergio Scarlatti. 2025-06-12. A note on D-functions and P-covariances on Hilbert spaces and related inequalities. https://arxiv.org/abs/2507.00009

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