I don't know how to prove the following statement:
Let $x, x'$ arbitrary and $x^*\in\partial h(x)$. If $h$ is $\rho$-strongly convex, i.e, $$h(\lambda x + (1-\lambda)x')\leq \lambda h(x) + (1-\lambda)h(x') - \rho \frac{\lambda(1-\lambda)} 2\|x-x'\|^2,\quad \forall x,x', \quad \lambda\in(0,1),$$ then: $$h(x)\ge h(x') + \left<x^*,x-x'\right> - \frac \rho 2 \|x-x'\|^2.$$