Inequalities Involving Berezin Norm and Berezin Number

We obtain new inequalities involving Berezin norm and Berezin number of bounded linear operators defined on a reproducing kernel Hilbert space H. Among many inequalities obtained here, it is shown that if A is a positive bounded linear operator on H, then ||A||(ber) = ber(A), where ||A||(ber) and ber(A) are the Berezin norm and Berezin number of A, respectively. In contrast to the numerical radius, this equality does not hold for selfadjoint operators, which highlights the necessity of studying Berezin number inequalities independently.