Let $\phi_1 \equiv \text{p.d.f of } \mathcal{N}(x; \mu_1, \sigma_1^2) $ and $\phi_2 \equiv \text{p.d.f of } \mathcal{N}(x; \mu_2, \sigma_2^2)$.
Further, let $\theta_1^* + \theta_2^* = 1$ where $\theta_i^* > 0$.
Also, let $A = \frac{\alpha_1(\alpha_1 + 1)}{(\alpha_1 + \alpha_1 + 1)(\alpha_1 + \alpha_2 + 2)}$, $B = \frac{\alpha_2(\alpha_2 + 1)}{(\alpha_1 + \alpha_2 + 1)(\alpha_1 + \alpha_2 + 2)}$ where $\alpha_i > 1$
Further, $A + B + C = 1$.
Under what conditions is $$\int_x \frac{\phi_1\phi_2(\theta_1^*\phi_1 + \theta_2^*\phi_2)}{A\phi_1^2 + B\phi_2^2 + C\phi_1\phi_2}dx \leq 1$$