I am proving an inequality. It looks like this $$\frac{(1-p_1)(1-r_2^Q)}{1-\delta r_2}>\frac{p_2(1-r_3^{-Q})}{1-\delta/r_3}~\Leftrightarrow~p_1+p_2<1,$$ where $$r_2=\frac{1-\sqrt{1-4\delta^2p_1(1-p_1)}}{2p_1\delta},~r_3=\frac{1-\sqrt{1-4\delta^2p_2(1-p_2)}}{2p_2\delta}$$ and $Q\geq 1$, $\delta\in(0,1)$, $p_1\in(1/2,1)$. and $p_2\in(0,1)$.
I've numerically verified that this claim is true. However, as the expression of $r_2,r_3$ is complicated and it is an if and only if statement. I had a hard time in proving it.