I have a function $f$ that is $L_1$-smooth, and that has $L_2$-Lipschitz continuous Hessian. Let the function $g$ be defined such that $g(x) = \nabla^2 f(x) x$. Under which conditions/assumptions, the function $g$ is Lipschitz continuous?
I tried the following: \begin{align} &\|g(x)-g(y)\|\\ &= \| \nabla^2 f(x) x- \nabla^2 f(y) y\|\\ &\leq \|\nabla^2 f(x)\| \|x-y\| + L_2\|x-y\|\|y\| \end{align}
I guess I can make the assumption that $\|\nabla^2 f(x)\| \leq M$ where $M$ is a constant but I'm having an issue bounding the second term.