Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a twice-differentiable convex function. It can shown be that for any $x_0 \in \mathbb{R}$:
$f(x) \leq f(x_0) + (x-x_0)f'(x_0) + \frac{(x-x_0)^2}{2}$C
where $C = \arg \max_x f''(x)$ is the maximum curvature of $f(x)$ over its domain. This result has been proven here:
Bohning, D. and Lindsay, B.G., ``Monotonicity of quadratic-approximation algorithms'', Annals of the Institute of Statistical Mathematics, vol. 40, no. 4, pp. 641-663, 1988.
Is it possible to derive a quadratic lower bound as well using a similar idea? It seems intuitively possible with a small enough choice of curvature in the bound.