A functional inequality related to Domar's uniform boundedness theorem

We study the functional inequality \[ f(r+s)\le g(r)+\alpha f(s) \quad(r,s>0). \] Here $g:(0,\infty)\to[0,\infty)$ is a given decreasing function, $\alpha$ is a constant such that $0<\alpha<1$, and the problem is to determine whether the family of decreasing functions $f:(0,\infty)\to[0,\infty)$ that satisfy this inequality is bounded above by some finite function on $(0,\infty)$ and, if so, to find bounds for this function.

We present a solution to this problem, and use it to give a new proof of a theorem of Domar on the uniform boundedness of certain families of subharmonic functions, in addition obtaining explicit bounds.