Generating $\tilde{s}$.

Let us define $V:=\omega$, $D:=H \omega$, $r:=\beta / \alpha$, equation 5.8, can now be rewritten:

$$\frac{(V + r D)^T H ( V + r D)}{(V+rD)^2} = \frac{V^T H V + r V^T H D + r D^T H V + r^2 D^T H D}{ V^2 + 2 r V^T D + r^2 D^2}$$

$$= \frac{r^2 D^T H D + 2 r V^T H D+ V^T H V}{ V^2 + 2 r V^T D+ r^2 D^2} = f(r)$$

We will now search for $r^*$, root of the Equation

$$\frac{ \partial f(r^*)}{\partial r} =0 \nonumber$$

$$\Leftrightarrow \quad ( 2 r D^T H D + 2 V^T H D) (V^2 + 2 r V^T D + r^2 D^2)$$

$$\quad - (r^2 D^T H D + 2 r V^T H D + V^T H V)(2 r D^2 + 2 V^T D) =0$$

$$\Leftrightarrow \quad \bigg[ (D^T H D) (V^T D) - D^2 D^2 \bigg] r^2 + \bigg[ (D^T H D) V^2 - D^2 (V^T D) \bigg] r + \bigg[ D^2 V^2 - (V^T D)^2 \bigg]=0$$

(Using the fact that $D=HV \Leftrightarrow D^T=V^T H^T=V^T H$.)
We thus obtain a simple equation $a x^2+b x+c=0$. We find the two roots of this equation and choose the one $r^*$ which maximize 5.8. $\tilde{s}$ is thus $V+ r^* D$.