Let <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> be an additive function and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">T</mi> </math></EquationSource> </InlineEquation> be the unit circle in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathbb R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>. W. Benz asked in 1990 whether the condition <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$xf(y)=yf(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for all points (x, y) in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">T</mi> </math></EquationSource> </InlineEquation> implies that f is linear, and whether the condition <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$xf(x) + yf(y) = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>y</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(x,y)\in \mathcal {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi mathvariant="script">T</mi> </mrow> </math></EquationSource> </InlineEquation> implies that f is a derivation. In 2005 Boros and Erdei showed that the answer to each question is affirmative. Here we answer similar questions when <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">T</mi> </math></EquationSource> </InlineEquation> is replaced by other conic sections.

Conditional equations for additive functions on conic sections

期刊论文
发表日期：

2026年2月16日

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Bruce Ebanks

摘要

Let $f:{\mathbb R}\rightarrow {\mathbb R}$ $f : R \to R$ be an additive function and let $\mathcal {T}$ $T$ be the unit circle in ${\mathbb R}^2$ $R^{2}$ . W. Benz asked in 1990 whether the condition $xf(y)=yf(x)$ $x f (y) = y f (x)$ for all points (x, y) in $\mathcal {T}$ $T$ implies that f is linear, and whether the condition $xf(x) + yf(y) = 0$ $x f (x) + y f (y) = 0$ for all $(x,y)\in \mathcal {T}$ $(x, y) \in T$ implies that f is a derivation. In 2005 Boros and Erdei showed that the answer to each question is affirmative. Here we answer similar questions when $\mathcal {T}$ $T$ is replaced by other conic sections.