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 $f$.\\
Deduce that for all $x, y \in \mathbb{R}$, denoting $\tilde{x} := x - \tau f'(x)$ and $\tilde{y} := y - \tau f'(y)$, we have
$$|\tilde{x} - \tilde{y}|^2 \leq |x-y|^2(1 - \alpha\tau(2 - L\tau)).$$