Davis-Wielandt-Berezin radius inequalities of Reproducing kernel Hilbert space operators

Several upper and lower bounds of the Davis-Wielandt-Berezin radius of bounded linear operators defined on a reproducing kernel Hilbert space are given. Further, an inequality involving the Berezin number and the Davis-Wielandt-Berezin radius for the sum of two bounded linear operators is obtained, namely, if $A$ and $B$ are reproducing kernel Hilbert space operators, then $$\eta(A+B) \leq \eta(A)+\eta(B)+\textbf{ber}(A^*B+B^*A),$$ where $\eta(\cdot)$ and $\textbf{ber}(\cdot)$ are the Davis-Wielandt-Berezin radius and the Berezin number, respectively.