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Let us consider that $f:\mathbb{R}\rightarrow\mathbb{R}$ is a smooth function and $A, B$ are positive constants.

Let us define the function $g(x)$ given by: $$ g(x) = -A f(x) + B f''(x). $$

I wonder about conditions of $f(x)$ that makes $g(x)$ bounded from below.

For example if $f(x)$ is bounded from above and convex then of course $g(x)$ is bounded from below. But I want to find more general rules.

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