Say that I am given an infinitely differentiable convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$.
I am wondering if I can construct a meaningful lower bound approximation of $f$ using it's minimum value $f(x^*)$ as well as the value of it's second derivative $\nabla^2 f(x^*)$ at the minimum.
Specifically, I am looking for a function $g: \mathbb{R}^n \rightarrow \mathbb{R}$ such that:
$$g(x) \leq f(x) \text{ for all } x \in \mathbb{R}^n$$
By meaningful, I mean any function $g(x)$ other than the constant function $g(x) = f(x^*)$.
EDIT: I have asked the question in a general form, but it if helps, the function that I am interested in is the $L_2$-regularized logistic loss function $$f(x) = \log(1+\exp(s^Tx)) + C\|x\|^2_2$$
where $s \in \mathbb{R}^n$ and $C > 0$.
The function is not only infinitely differentiable in $x$, but is also strictly convex in $x$ and with a unique minimum.
EDIT #2: To add to @Michael's answer, you can also find a non-trivial lower / upper bound for strictly convex functions in Section 9.1.2 of Stephen Boyd's Convex Optimization Book.