Given a bounded open subset <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> and closed subsets A, B of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^k$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation>, we discuss when an estimate <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$u(x)\le g(\operatorname {dist}(x,A\cup B))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mo>dist</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$x\in \Omega \setminus (A\cup B)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, for a function u subharmonic on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Omega \setminus B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation>, implies that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$u(x)\le h(\operatorname {dist}(x,B))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mo>dist</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$x\in \Omega \setminus B$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$g,h:(0,\infty )\rightarrow (0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>,</mo> <mi>h</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are decreasing functions and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$g(0^+)=h(0^+)=\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. 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.

Self-improving Estimates of Growth of Subharmonic and Analytic Functions

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2026年2月17日

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Glenier Bello,
Dmitry Yakubovich

摘要

Given a bounded open subset $\Omega $ $Ω$ and closed subsets A, B of $\mathbb {R}^k$ $R^{k}$ , we discuss when an estimate $u(x)\le g(\operatorname {dist}(x,A\cup B))$ $u (x) \leq g (dist (x, A \cup B))$ , $x\in \Omega \setminus (A\cup B)$ $x \in Ω \ (A \cup B)$ , for a function u subharmonic on $\Omega \setminus B$ $Ω \ B$ , implies that $u(x)\le h(\operatorname {dist}(x,B))$ $u (x) \leq h (dist (x, B))$ , $x\in \Omega \setminus B$ $x \in Ω \ B$ , where $g,h:(0,\infty )\rightarrow (0,\infty )$ $g, h : (0, \infty) \to (0, \infty)$ are decreasing functions and $g(0^+)=h(0^+)=\infty $ $g (0^{+}) = h (0^{+}) = \infty$ . We seek for explicit expressions of h in terms of g. We give some results of this type and show that Domar’s work 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.