Structure of quadratic inequalities

When A and B are n × n positive semi-definite matrices, and C is an n × n Hermitian matrix, the validity of a quadratic inequality $\text{(x}{}^1\text{Ax)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(x}{}^1\text{Bx)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{? ¦x}{}^1\text{Cx¦}$ is shown to be equivalent to the existence of an n × n unitary matrix W such that $\text{A}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{WB}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+ B}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{W}{}^1\text{A}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 2C.}$ Some related inequalities are also discussed.