β-smoothness

A function $f$ is $\beta$ smooth if, for all $\mathbf{x},\mathbf{y}$: $$||\nabla f(\mathbf{x})-\nabla f(\mathbf{y})||_2 \leq \beta ||\mathbf{x}-\mathbf{y}||_2$$

For scalar valued function $f$, equivalent to $f''(x) \leq \beta$.

For scalar functions, a twice-differentiable function $f$ is α-strongly convex and $\beta$-smooth if for all $x$, $$\alpha \leq f''(x) \leq \beta$$

References:

1. https://arxiv.org/pdf/1405.4980.pdf