<p>In the paper we improve some results from [<CitationRef CitationID="CR9">9</CitationRef>] concerning the Goła̧b–Schinzel inequality <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f(x+f(x)y)\le f(x)f(y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for a continuous unknown function <InlineEquation ID="IEq2"> <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>. We also show some properties of solutions of the generalized inequality <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f(x+g(x)y)\le f(x)f(y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for continuous unknown functions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$f,g:\mathbb {R}\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> provided <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$g\ge f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>≥</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

On continuous solutions of a generalized Goła̧b–Schinzel functional inequality

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2026年5月16日

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Eliza Jabłońska,
Zuzanna Żurek

摘要

In the paper we improve some results from [9] concerning the Goła̧b–Schinzel inequality $f(x+f(x)y)\le f(x)f(y)$ $f (x + f (x) y) \leq f (x) f (y)$ for a continuous unknown function $f:\mathbb {R}\rightarrow \mathbb {R}$ $f : R \to R$ . We also show some properties of solutions of the generalized inequality $f(x+g(x)y)\le f(x)f(y)$ $f (x + g (x) y) \leq f (x) f (y)$ for continuous unknown functions $f,g:\mathbb {R}\rightarrow \mathbb {R}$ $f, g : R \to R$ provided $g\ge f$ $g \geq f$ .