Some new relations between the Berezin number and the Berezin norm of operators

We prove new Grüss type inequalities for the Berezin symbol of some operator products. As biproducts, we find new relations between the Berezin number, the Berezin norm and some new quantities for operators acting on the reproducing kernel Hilbert space. In particular, we prove for arbitrary bounded linear operator A that ber(A2)≤14((ber(A4))+∥A2∥Ber2), where ber(⋅) and ∥⋅∥Ber denote, respectively, the Berezin number and the Berezin norm of operator A.