Given the function
$g(n,m)=\min\Big\{f(a,b)+f(n-a,c)+f(n,m-bc)\Big|\\a,b,c\ \ \text{with} \left\{\begin{matrix} a,\ b,\ n-a,\ c,\ m-bc \geq 0 \\ b\leq a! \\ c\leq (n-a)! \\ \end{matrix}\right. \Big\} $
Assuming that $n,m\geq 0,\ ((\lceil n/2\rceil)!)^2\leq m\leq n!,\ f(n,m)=\Omega (n)$,
is it true that $g(n,m) \geq 2f(\lfloor n/2\rfloor ,(\lfloor n/2\rfloor)!)+f(n,m-((\lceil n/2\rceil)!)^2)$ ?
I tried KKT conditions, but can't derive this (as it contains factorial).
Also, it seems that the condition $f(n,m)=\Omega (n)$ implies that $f$ is convex on our domain (and thus, satisfies the regularity condition for using KKT), but I managed to prove it only if $f$ is polynomial.
So I am fully stuck in this...
Any help would be highly appreciated!