We are given $f \in \mathcal{C}^1(\mathbb{R})$ such that $f'$ is $L$-Lipschitzian, with $L > 0$, $f$ is $\alpha$-convex with $\alpha > 0$, and $x_*$ denotes a minimizer of $f$.\\
Show that for all $x, y \in \mathbb{R}$
$$\alpha|x-y|^2 \leq \left(f'(x) - f'(y)\right)(x-y)$$