Berezin radius inequalities via unified convexity approach: a general framework encompassing classical results

We present a unified approach to Berezin radius inequalities based on weighted power mean inequalities and convex analysis. Our main contribution is a general interpolation framework showing that classical inequalities, including Cauchy-Schwarz, Buzano, Kato’s mixed Schwarz inequality, and their generalizations, emerge as special cases of a single master inequality. We demonstrate that the results in the recent paper by Saltan, Gürdal, Garayev, and Rajawat [22] can be understood and generalized through our unified approach. Moreover, we establish bounds with optimal parameters and derive new inequalities, including variational characterizations and dual formulations.