  Consider $\mathcal{H}$ is a complex Hilbert space and $A$ is a positive
operator on $\mathcal{H}.$ The mapping $\langle\cdot,\cdot\rangle_A:
\mathcal{H}\times \mathcal{H} \to \mathbb {C}$, defined as $\left\langle
y,z\right\rangle_{A}=\left\langle Ay,z\right\rangle $ for all $y,z$ $\in $
${\mathcal{H}}$, induces a seminorm $ \left\Vert \cdot\right\Vert_{A}$. The
$A$-Davis-Wielandt radius of an operator $S$ on $\mathcal{H}$ is defined as
$d\omega_{A}\left( S\right) =\sup \left\{ \sqrt{\left\vert \left\langle
Sz,z\right\rangle_{A}\right\vert ^{2}+\left\Vert Sz\right\Vert_{A}^{4}}
:\left\Vert z\right\Vert_{A}=1\right\} \text{.} $ We investigate some new
bounds for $d\omega_{A}\left( S\right)$ which refine the existing bounds. We
also give some bounds for the $2\times 2$ off-diagonal block matrices.
