Is there any property that captures $F(\alpha x) \geq \beta F(x)$ for $1>\alpha,\beta>0?$

Question

Given a CDF $F:\mathbb{R}_{\geq 0}\to [0,1]$ such that $F(\alpha x) \geq \beta F(x)$ for some $1>\alpha,\beta>0.$ I'm trying to figure out if there is any known family of distributions for which this condition holds. Is this somehow related to log-concavity? Is there any connection to reverse hazard rate?