Abstract
We consider operator A on the reproducing Kernel Hilbert space H = H (Omega) over some set Omega with the reproducing kernel K-lambda (z) = K (z, lambda) and define Davis-Wielandt-Berezin radius eta(A) by the formula
eta(A): = sup {root vertical bar(A) over tilde(lambda)vertical bar(2) + parallel to AK(lambda)parallel to(4) : lambda is an element of Omega},
where (A) over tilde is the Berezin symbol of A defined by (A) over tilde(lambda) := < A (K-lambda) over cap, (K-lambda) over cap > lambda is an element of Omega, where (K-lambda) over cap > lambda = K-lambda/parallel to K-lambda parallel to(H) is the normalized reproducing kernel of H. We prove several inequalities for this new quantity eta(A) involving known Dragomir inequalities. Some other Berezin number inequalities are also proved.