Abstract
We give in terms of Berezin symbols some refinements of Holder-McCarthy inequality and Young inequality for positive operators on the reproducing kernel Hilbert space. By applying these inequalities we prove some new estimates for the Berezin number of operators. We also discuss the power inequalities ber (A(n)) <= cber (A)(n) and ber <= Cber (A(n)) for integers n >= 1, which is not well studied yet.