A functional inequality related to Domar's uniform boundedness theorem

We study the functional inequality $$f(r+s)\le g(r)+\alpha f(s)(r,s>0)\mathrm{.}$$ Here $g:(0,\mathrm{\infty })\to [0,\mathrm{\infty })$ 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 $f:(0,\mathrm{\infty })\to [0,\mathrm{\infty })$ that satisfy this inequality is bounded above by some finite function on $(0,\mathrm{\infty })$ 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.