Where here we prove an approximation bound for twice continuously differentiable functions with M-Lipschitzian Hessian, that is for all . In particular, we show that for all
This is stated as Lemma 1.2.4 in: Y. Nesterov, Introductory Lectures on Convex Optimization – A basic course, Kluwer Ac. Publishers, 2004.
Since is twice continuously differentiable, we may use the fact that
for some .
We may write this equivalently as (and this is maybe more convenient)
Since is -Lipschitz,
where . We may now easily plug (3) into (2) and prove the original inequality. In particular, because of (1):
where the first integral in (4) is
and the second integral’s absolute value is