Given $0<\alpha, p, q<1$, let, $$ C=1-2[\alpha(1-p)d_{0} +(1-\alpha)(1-q)(1-d_{0})+\alpha p d_{1} +(1-\alpha)q(1-d_{1})] $$
where,
$D_{0}=\sqrt{\frac{p}{1-p}}-\sqrt{\frac{q}{1-q}}-log_{2}\frac{\alpha}{1-\alpha}$
$D_{1}=\sqrt{\frac{1-p}{p}}-\sqrt{\frac{1-q}{q}}-log_{2}\frac{\alpha}{1-\alpha}$
$d_{k}=1$ if $D_{k}>0$ $and$ $d_{k}=0$ if $D_{k}\le0$
I want to prove that $C\ge0$ for all possible $\alpha$, $p$, $q$.