Given $F(\underline{x}) = Ax_1 + Bx_2 + \ln(a^2-(x_1^2+x^2_2))$ on $S=\{\underline{x}\in\mathbb{R}\mid x_1^2+x_2^2<a^2\}$ with $A,B,a\in\mathbb{R}$, show that $F$ is concave on $S$.
Since we have to prove this for the set $S$, a simple second derivative may not work. I am trying to solve this using the condition below but if there any other ways to solve it, please let me know
$$f(\lambda x_1+(1-\lambda)x_2)\geq \lambda f(x_1)+(1-\lambda)f(x_2)\quad\forall\lambda\in[0,1]$$
Any ideas on how to prove this?
You can simply use standard composition rules you can find for instance on the textbook
http://www.stanford.edu/~boyd/cvxbook/
pag 84-86 and example 3.13. $Ax_1+Bx_2$ is affine and hence concave. The natural logarithm and its argument are both concave functions either. On $S$ the logarithm argument is strictly positive. Then
concave + concave + concave(strictly positive concave) = concave
please check the details.