A functional inequality related to Domar's uniform boundedness theorem

We study the functional inequality \[ f(r+s)\le g(r)+αf(s) \quad(r,s>0). \] Here $g:(0,\infty)\to[0,\infty)$ is a given decreasing function, $α$ is a constant such that $0<α<1$, and the problem is to determine whether the family of decreasing functions $f:(0,\infty)\to[0,\infty)$ that satisfy this i