Self-improving estimates of growth of subharmonic and analytic functions

Given a bounded open subset $\mathrm{\Omega }$ and closed subsets $A,B$ of ${\mathbb{R}}^k$, we discuss when an estimate $u(x)\le g(dist(x,A\cup B))$, $x\in \mathrm{\Omega }\setminus (A\cup B)$, for a function $u$ subharmonic on $\mathrm{\Omega }\setminus B$, implies that $u(x)\le h(dist(x,B))$, $x\in \mathrm{\Omega }\setminus B$, where $g,h:(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ are decreasing functions and $g(0^{+})=h(0^{+})=\mathrm{\infty }$. We seek for explicit expressions of $h$ in terms of $g$. We give some results of this type and show that Domar’s work (On the existence of a largest subharmonic minorant of a given function, Ark. Mat., 3 (1957), pp. 429-440) permits one to deduce other results in this direction. Then we compare these two approaches. Similar results are deduced for estimates of analytic functions.