I want to show that $t(a,b)$ where $0\leq a\leq 1,~0\leq b\leq 1$ and $p,q>0$ is as a concave/convex function: $$t(a,b)=\frac{\left(\frac{1-b}{2-b}\right) \left(\frac{2 a}{1-a}+b\right)}{p\left(\frac{2 a}{1-a}+b\right)+q\left(\frac{1-b}{2-b}\right)+1}$$
Although the function is twice differentiable, it is really difficult to discuss the Hessian properties since it is ridiculously complected expression.
Is there any alternative way?
You have $t(0,b)=\frac{b-b^2}{-pb^2+(2p-q-1)b+q+2}$. This expression is neither convex nor concave (draw a plot for p=q=1). Including $a$ as a variable will only make things worse.