Estimation of $\mathrm{A}$-Berezin number and $\mathrm{A}$-Berezin norm inequalities via Moore-Penrose inverse

In this article, we establish the $A$-Berezin number and $A$-Berezin norm inequalities for bounded linear operators on a reproducing kernel Hilbert space using the Moore-Penrose inverse. We further extend these inequalities to the case of the sum of two bounded linear operators. As an application, upper bounds of the Berezin number and Berezin norm of block matrices have been discussed via the Moore-Penrose inverse. The inequalities established here offer both refinements and generalizations of previous results.