On the Hilbert space $ H $, $ A $ is a non-negative self-adjoint operator and $ B $ is a symmetric operator. Let $ D(B)\supset D(A) $, where $ D(A) $ and $ D(B) $ are definite domain for $ A $ and $ B $ repectively. Assume that \begin{align*} \|Bx\|\leq \|Ax\|,\quad\forall x\in D(A). \end{align*} Show that $ |(Bx,x)|\leq (Ax,x) $ for any $ x\in D(A) $.
Here I want to use the norm $ \|\cdot\| $ to represent the $ (Ax,x) $ and $ (Bx,x) $ but I do not how to go on. Can you give me some hints or references?