A functional inequality related to Domar’s uniform boundedness theorem

Abstract

We study the functional inequality

$$ f(r+s)\le g(r)+\alpha f(s) \quad (r,s>0). $$

Here \(g:(0,\infty )\rightarrow [0,\infty )\) 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 \(f:(0,\infty )\rightarrow [0,\infty )\) 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, while also obtaining explicit bounds.