Self-improving Estimates of Growth of Subharmonic and Analytic Functions

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