Consider the following function,
$ f(x, y) = e^{m e^{-y}+n e^{-x}-x-y} \left(a x e^y+b e^x y+c x y\right) $
where $a, b, c, m$ and $n$ are positive constants.
I want to show $f(x, y)$ is quasi-concave. To that end, we define the set $S_{\alpha} \subset \mathbb R^2$ by,
$ S_{\alpha} = \{ (x, y) | f(x, y) > \alpha \} $
How can I show the set $S_{\alpha}$ is convex? Is there any simpler way than using the definition $f(x_1, y_1) > \alpha \wedge f(x_2, y_2) > \alpha \Rightarrow f(t x_1+(1-t)x_2, t y_1+(1-t)y_2) > \alpha \;\text{ for }\; 0 < t < 1?$ Is it possible to show convexity by composing some basic functions like $xy$, $x e^y$, ...?