The following function seems to be concave (I could not prove it) $$r(a,b)=\frac{1}{2} (1-a) \log \left(p \left(\frac{2 a}{1-a}+b\right)+1\right)$$ where $a\in[0,1)$, $b\in[0,1)$ and $p>0$. For $p=0.45$:
When corresponding eigenvalues of the Hessian matrix are plotted, they are positive and negative:
However, the function is concave if Hessian is negative semidefinite. So I am confused with this observation.
How can I prove that $r(a,b)$ convex/concave?