This work presents new insights into the behavior of Berezin numbers for bounded linear operators on reproducing kernel Hilbert spaces (RKHS). Focusing on operators that admit a Moore Penrose inverse, we derive a variety of refined inequalities that extend classical Berezin-type bounds. Our approach incorporates generalized mean functions, convexity techniques, and operator-theoretic tools to establish tighter upper estimates involving both the operator and its generalized inverse. The analysis further employs interpolational methods and positivity of block operator matrices to sharpen known results and produce novel estimates. Additionally, we utilize structural properties of doubly convex functions and variants of inner product inequalities, including those inspired by the Buzano and Schwarz inequalities. The results offer a unified framework for comparing Berezin numbers, numerical radii, and related quantities, with potential applications in operator theory and functional analysis.
Ren et al. (Wed,) studied this question.