Let T be a bounded linear operator on a complex Hilbert space and d w(T) denote the Davis–Wielandt radius of the operator T. We prove that dw2(T)≤min0≤α≤1‖α|T|2+(1-α)|T∗|2+|T|4‖and dw2(T)≤min0≤α≤1‖α|T|2+(1-α)|T∗|2+|T∗|4‖,where |T|=T∗T,|T∗|=TT∗. We also develop several other bounds for the Davis–Wielandt radius and prove that the bounds obtained here are better than the existing ones.