Reversed and Refined Berezin Number Inequalities via Buzano’s Inequality and Polar Decomposition

The Berezin number and Berezin norm have become indispensable tools for analyzing the structure of operators on reproducing kernel Hilbert spaces, yet the gap between these quantities and the operator norm is still far from fully understood. In this paper, we narrow that gap by establishing several new reverse-type inequalities for ber(V) and ∥V∥ber˜ under natural disc-localization and accretivity conditions. Combining a refinement of the Schwarz inequality due to Dragomir with Buzano’s inequality, we further derive sharp bounds for ber(A*TA) and for products TA, together with their power-type extensions. Applying the polar decomposition, we also obtain companion inequalities for |T|α and the Aluthge transform. Finally, for operators with closed range, we use the Moore–Penrose inverse to derive new reverse-type Berezin inequalities that refine and complement the results obtained via the polar decomposition, and we illustrate the sharpness of these bounds with an explicit example.