<p>The present paper is devoted to the functional inequality <Equation ID="Equ15"> <EquationSource Format="TEX">$ f(x+y) + f(y+z) + f(x+z) \le f(x+y+z) + f(x) + f(y) + f(z) $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </Equation>for the unknown mapping <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f:\mathbb {R}\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, which is satisfied for almost all <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(x,y,z)\in \mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> (i.e., for all <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(x,y,z)\in \mathbb {R}^3\setminus M$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M\subset \mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> is a null set).</p>

Hlawka’s functional inequality almost everywhere

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2025年7月28日

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Włodzimierz Fechner,
Eliza Jabłońska

摘要

The present paper is devoted to the functional inequality $ f(x+y) + f(y+z) + f(x+z) \le f(x+y+z) + f(x) + f(y) + f(z) $ $f (x + y) + f (y + z) + f (x + z) \leq f (x + y + z) + f (x) + f (y) + f (z)$ for the unknown mapping $f:\mathbb {R}\rightarrow \mathbb {R}$ $f : R \to R$ , which is satisfied for almost all $(x,y,z)\in \mathbb {R}^3$ $(x, y, z) \in R^{3}$ (i.e., for all $(x,y,z)\in \mathbb {R}^3\setminus M$ $(x, y, z) \in R^{3} \ M$ , where $M\subset \mathbb {R}^3$ $M \subset R^{3}$ is a null set).