In this article, the concept of the A-Davis-Wielandt Berezin number is introduced for positive operator A. Some upper and lower bounds for the A-Davis-Wielandt Berezin number are proved. Moreover, some inequalities related to the concept of the Davis-Wielandt Berezin number are obtained, which are generalizations of known results. Among them, it is shown that ber 2 dw (S) ≤ inf γ∈C { 2||Re(γ)Re(S) + Im(γ)Im(S)|| + ||S * S − 2Re(¯ γS)|| 2 + 2||Re(¯ γS)|| − |γ| 2 + ber 2 (S − γI)}, where S ∈ B(H(Ω)). Also, we determined the exact value of the A-Davis-Wielandt Berezin number of some special type of operator matrices.