A-Davis-Wielandt-Berezin radius inequalities

We consider operator $V$ on the reproducing kernel Hilbert space $\mathcal{H}=\mathcal{H}(\Omega)$ over some set $\Omega$ with the reproducing kernel $K_{\mathcal{H},\lambda}(z)=K(z,\lambda)$ and define A-Davis-Wielandt-Berezin radius $\eta_{A}(V)$ by the formula $\eta_{A}(V):=sup\{\sqrt{| \langle Vk_{\mathcal{H},\lambda},k_{\mathcal{H},\lambda} \rangle_{A}|^{2}+\|Vk_{\mathcal{H},\lambda}\|_{A}^{4}}:\lambda \in \Omega\}$ and $\tilde{V}$ is the Berezin symbol of $V$ where any positive operator $A$-induces a semi-inner product on $\mathcal{H}$ is defined by $\langle x,y \rangle_{A}=\langle Ax,y \rangle$ for $x,y \in \mathcal{H}.$ We study equality of the lower bounds for A-Davis-Wielandt-Berezin radius mentioned above. We establish some lower and upper bounds for the A-Davis-Wielandt-Berezin radius of reproducing kernel Hilbert space operators. In addition, we get an upper bound for the A-Davis-Wielandt-Berezin radius of sum of two bounded linear operators.