Let us consider the following function: $$ p(x) = a f'(x) + bf''(x) - cf(x), $$ where $a, b, c$ are some positive constants.
I want to find the assumptions for $f(x)$ such that $p(x)$ function is bounded from below.
I can simply note that if $f'(x), f''(x)$ are positive and $f(x)$ is negative, then $p(x)$ is bounded below.
But I want to find more general rules for $f(x)$ for which $p(x)$ is bounded from below.
I suppose for example that for some concave $f(x)$ functions ($f''(x)<0$), the function $p(x)$ can be bounded from below.