So let's say I have the inner product:
$$\vec{y}_1^H \vec{y}_2 = (\vec{x}\circ\vec{h}_1)^{H} (\vec{x}\circ\vec{h}_2) = \sum_{i} (x_i^{\ast} h_{1,i}^*) (x_i h_{2,i}) = \sum_{i} |x_i|^2 h_{1,i}^* h_{2,i}$$
Is it true that I can say, by Hoelder's Inequality, that:
$$|\vec{y}_1^H \vec{y}_2| \le ||\vec{x}||^2_2 |\vec{h}_1^H \vec{h}_2|$$
I am fairly certain that if $\vec{h}_1$ and $\vec{h}_2$ are orthogonal, then $\vec{y}_1$ and $\vec{y}_2$ should be as well. But is this the right way to go about showing that?