I am trying to work out some stability conditions for ODE methods and during the computations one needs to solve the following inequality:
Let $\alpha_1, \alpha_2, \beta_1, \beta_2 \in \mathbb{R}$. For which $z \in \mathbb{C}$, $z=a+bi$ the following inequality holds
$$ |\alpha_2 z^2 + \alpha_1 z + 1| < | \beta_2 z^2 + \beta_1z + 1| .$$
I was able to reformulate the above into
$$ (\alpha_1^2 - \beta_1^2)(a^2+b^2) + (\alpha_2^2 - \beta_2^2)(a^2+b^2)^2 + 2\left( (\alpha_1 \alpha_2 - \beta_1 \beta_2)(a^2 +b^2)a + (\alpha_1 - \beta_1)a + (\alpha_2 - \beta_2)(a^2 -b^2) \right)<0.$$
I would be grateful for a push in any direction - especially a more elegant one than the 'brute force' I used above.