Consider H is a complex Hilbert space and A is a positive operator on H. The mapping ⟨·,·⟩A:H×H→C, defined as y,zA=Ay,z for all y, z∈H, induces a seminorm ·A. The A-Davis–Wielandt radius of an operator S on H is defined as dωAS=supSz,zA2+SzA4:zA=1. We investigate some new bounds for dωAS which refine the existing bounds. We also give some bounds for the 2×2 off-diagonal block matrices.