Upper bounds for Berezin numbers of self-adjoint operators and applications

We prove upper bounds for Berezin symbols and Berezin numbers of operators, with a special emphasis on the quadratic weighted operator geometric mean. In particular, we prove some upper bounds for Berezin symbols and Berezin numbers of some self-adjoint operators on reproducing kernel Hilbert spaces over some suitable sets. Moreover, we estimate the best possible constant in some generalized Hardy-Hilbert inequality for the series.