New estimates for the Berezin number and norm of operators via Buzano and Furuta-type inequalities

Suppose that (KF, · , · ) represents a reproducing kernel Hilbert space defined over a non-empty set F. Let ξ κ and ξ υ designate the normalized reproducing kernels within KF. For a bounded linear operator U acting on KF, its Berezin number and Berezin norm are defined, respectively, by $$ ber(U) = κ ∈ F | U {ξ }κ, {ξ }κ | and \|U\|ber = κ , υ ∈ F | U {ξ }κ, {ξ }υ |. $$ In this paper, we establish new upper bounds for these fundamental quantities. Our approach relies on several refined Buzano-type inequalities together with related Cauchy-Schwarz-type estimates. Moreover, we develop a class of Furuta-type inequalities for bounded linear operators by combining mixed Schwarz-type techniques with spectral radius arguments. These operator inequalities are then applied to derive several refined bounds for the Berezin number and Berezin norm, including estimates for operator sums and products as well as Hölder-type inequalities. The obtained results extend and complement various known inequalities in the recent literature.