New inequalities for some quadratic forms and related results

We prove some new Jensen type inequalities for the Berezin symbol of self-adjoint operators and some class of positive operators on the reproducing kernel Hilbert spaceH(?) over some set ?. Recall that the Berezin symbol ? of operator A on H(?) is defined by the following special type of quadratic form: ?(?) := ?A?k?,?k??, ? ? ?, where k? is the reproducing kernel of the space H(?), i.e., f (?) = ?f,?k?? for all f ? H(?) and ? ? ?;?k? := k?/||k?||H(?)is the normalized reproducing kernel.