I am stuck with proving the following inequality. I am to show this:
$Y \leq -2 \sigma \delta$
starting with these:
Eq.1
$Y = -2 \sigma \left[ \rho x^2 + y^2 + \beta (z - \rho)^2 - \beta \rho^2 \right]$
Eq. 2 (how it was written)
$\rho + \sqrt{\rho^2 + \delta/\beta} \leq \left(z - 2 \rho \right)^2$
Eq. 2 (how I interpret it since it was written this way)
$\rho + \sqrt{\rho^2 + \frac{\delta}{\beta}} \leq \left(z - 2 \rho \right)^2$
This is what I have so far:
From Eq. 2
$\rho + \sqrt{\rho^2 + \frac{\delta}{\beta}} \leq \left(z - 2 \rho \right)^2$
$\sqrt{\rho^2 + \frac{\delta}{\beta}} \leq \left(z - 2 \rho \right)^2 - \rho$
$\rho^2 + \frac{\delta}{\beta} \leq \left(z - 2 \rho \right)^4 - 2 \rho \left(z - 2 \rho \right)^2$
$\delta \leq \beta \left(z - 2 \rho \right)^4 - 2 \beta \rho \left(z - 2 \rho \right)^2$
From Eq. 1
$-2 \sigma \left[ \rho x^2 + y^2 + \beta \left(z - \rho \right)^2 - \beta \rho^2 \right] = -2 \sigma \left[ \rho x^2 + y^2 + \beta \left(z^2 - 2 \rho z \right) \right]$
$-2 \sigma \left[ \rho x^2 + y^2 + \beta \left(z - \rho \right)^2 - \beta \rho^2 \right] = -2 \sigma \left[ \rho x^2 + y^2 + \beta \left(z^2 - 2 \rho \right)^2 + 2 \beta \rho \left(z - 2 \rho \right) \right]$
$-2 \sigma \left[ \rho x^2 + y^2 + \beta \left(z - \rho \right)^2 - \beta \rho^2 \right] = -2 \sigma \left[ \rho x^2 + y^2 + \beta \left(z^2 - 2 \rho \right)^2 - 2 \beta \rho \left(z - 2 \rho \right) + 4 \beta \rho \left(z - 2 \rho \right) \right]$
This is where I am stuck. I don't know how to apply my inequality from Eq. 2 into Eq. 1. The thing that is weird is the degree in Eq. 2 differs from the degree in Eq. 1 (4 vs. 2, respectively).
I feel like there might be something wrong with the given Eq. 2. Maybe it shouldn't have a square root, or $\rho$ outside of the square root should be to the degree of 2, or maybe what is inside the square root is not written properly in the paper. It could be the following:
$\rho^2 + \sqrt{\rho^2 + \frac{\delta}{\beta}}$ or
$\rho + \sqrt{\frac{\rho^2 + \delta}{\beta}}$ or
$\rho + \rho^2 + \frac{\delta}{\beta}$ and many more.