This work extends the classical Grüss inequality from Euclidean disks in \(\mathbb {R}^2\) to the generalized p-disks within Minkowski space, by constructing a formulation that rigorously holds over these non-Euclidean domains. This extension establishes a refined framework of the Grüss inequality within the broader geometry of Minkowski spaces. Specifically, for functions defined on p-disks that meet certain Hölder-type conditions, several Grüss-type inequalities are established. As an application, we derive bounds for the covariance of random variables defined by a probability density function or a joint distribution.
