Let $\vec{u}$ and $\vec{v}$ be two vectors in $\mathbb{R}^{2}$ with $\vec{u} \cdot \vec{v} \geq 0$. Let $u = |\vec{u}|$ and $v = |\vec{v}|$. Is it possible to prove that $$\frac{u^{2}}{4}(1 - \frac{\vec{u} \cdot \vec{v})}{v^{2}})(v^{2} - 2 \vec{u} \cdot \vec{v})^{2} + \frac{v^{2}}{4}(1 - \frac{\vec{u} \cdot \vec{v}}{u^{2}})(u^{2} - 2 \vec{u} \cdot \vec{v})^{2} + (\vec{u} \cdot \vec{v})^{3}(\frac{u^{2}}{v^{2}} + \frac{v^{2}}{u^{2}} - 2) \geq 0$$ ?
It is clear that $\frac{u^{2}}{v^{2}} + \frac{v^{2}}{u^{2}} \geq 2$ because this is equivalent to $u^{4} + v^{4} \geq 2u^{2}v^{2}$. Is it possible to show that
$$\frac{u^{2}}{4}(1 - \frac{\vec{u} \cdot \vec{v})}{v^{2}})(v^{2} - 2 \vec{u} \cdot \vec{v})^{2} + \frac{v^{2}}{4}(1 - \frac{\vec{u} \cdot \vec{v}}{u^{2}})(u^{2} - 2 \vec{u} \cdot \vec{v})^{2} \geq C_{1}(u^{2} - 2\vec{u}\cdot \vec{v})^{2}$$ and/or $$\frac{u^{2}}{4}(1 - \frac{\vec{u} \cdot \vec{v})}{v^{2}})(v^{2} - 2 \vec{u} \cdot \vec{v})^{2} + \frac{v^{2}}{4}(1 - \frac{\vec{u} \cdot \vec{v}}{u^{2}})(u^{2} - 2 \vec{u} \cdot \vec{v})^{2} \geq C_{2}(v^{2} - 2\vec{u}\cdot \vec{v})^{2}$$
for positive quantities $C_{1}$ and $C_{2}$ (depending on $\vec{u}$ and $\vec{v}$ of course if necessary)?
I suspect that $$\frac{u^{2}}{4}(1 - \frac{\vec{u} \cdot \vec{v})}{v^{2}})(v^{2} - 2 \vec{u} \cdot \vec{v})^{2} + \frac{v^{2}}{4}(1 - \frac{\vec{u} \cdot \vec{v}}{u^{2}})(u^{2} - 2 \vec{u} \cdot \vec{v})^{2} + (\vec{u} \cdot \vec{v})^{3}(\frac{u^{2}}{v^{2}} + \frac{v^{2}}{u^{2}} - 2) = 0$$
if only if $u^{2} = v^{2} = 2\vec{u} \cdot \vec{v}$. Can this argument also be extended to vectors in $\mathbb{R}^{n}$ for arbitrary $n$?
Note: This expression should be equivalent to $$\frac{1}{4}u^{2}v^{2}(u^{2} + v^{2} - 10\vec{u} \cdot \vec{v}) + 2(\vec{u} \cdot \vec{v})^{2}(u^{2} + v^{2} - \vec{u} \cdot \vec{v})$$ So, if anyone can also help me show that
$$\frac{1}{4}u^{2}v^{2}(u^{2} + v^{2} - 10\vec{u} \cdot \vec{v}) + 2(\vec{u} \cdot \vec{v})^{2}(u^{2} + v^{2} - \vec{u} \cdot \vec{v}) \geq 0$$
and equality holds if and only if $u^{2} = v^{2} = 2 \vec{u} \cdot \vec{v})$ (as I suspect), it would be greatly appreciated. Thanks!