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

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.