Two functional equations preserving functional forms
Authors: Aczél János , Luce R. Duncan 
Published: July 2002

Two functional equations are considered that are motivated by three considerations: work in utility theory and psychophysics, questions concerning when pairs of degree 1 homogeneous functions can be homomorphic and calculating their homomorphisms, and the link of the latter questions to quasilinear mean values. The first equation is

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\begin{equation*}\boldsymbol{{\mathit{h}}}(\boldsymbol{{\mathrm{{\sigma}}}}(\boldsymbol{{\mathit{y}}})\boldsymbol{{\mathit{x}}{\mathrm{\hspace{.167em}+\hspace{.167em}}}}[\boldsymbol{{\mathrm{1\hspace{.167em}-\hspace{.167em}{\sigma}}}}(\boldsymbol{{\mathit{y}}})]\boldsymbol{{\mathit{y}}})\boldsymbol{{\mathrm{\hspace{.167em}=\hspace{.167em}{\tau}}}}(\boldsymbol{{\mathit{y}}})\boldsymbol{{\mathit{h}}}(\boldsymbol{{\mathit{x}}})\boldsymbol{\hspace{.167em}}\end{equation*}\end{document}

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\begin{equation*}\boldsymbol{{\mathrm{+\hspace{.167em}}}}[\boldsymbol{{\mathrm{1\hspace{.167em}-\hspace{.167em}{\tau}}}}(\boldsymbol{{\mathit{y}}})]\boldsymbol{{\mathit{h}}}(\boldsymbol{{\mathit{y}}})\boldsymbol{\hspace{1em}}(\boldsymbol{{\mathit{x}}{\mathrm{\hspace{.167em}{\geq}\hspace{.167em}}}{\mathit{y}}{\mathrm{\hspace{.167em}{\geq}\hspace{.167em}0}}})\boldsymbol{{\mathrm{,}}}\end{equation*}\end{document}

where
h
maps [0, ∞[ into a subset of [0, ∞[ and is strictly increasing and continuously differentiable; the functions σ and τ map [0, ∞[ continuously into [0, 1], σ(
y
) > 0 for
y
> 0 but σ is not 1 on ]0, ∞[. The solutions are fully determined. (Recently Zsolt Páles has eliminated the differentiability assumption.) The second equation is

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\begin{equation*}\boldsymbol{{\mathit{h}}}[\boldsymbol{{\mathit{y}}{\mathrm{\hspace{.167em}+\hspace{.167em}}}{\mathit{f}}}(\boldsymbol{{\mathit{x}}{\mathrm{\hspace{.167em}-\hspace{.167em}}}{\mathit{y}}})]\boldsymbol{{\mathrm{\hspace{.167em}=\hspace{.167em}}}{\mathit{h}}}(\boldsymbol{{\mathit{y}}})\boldsymbol{{\mathrm{\hspace{.167em}+\hspace{.167em}}}{\mathit{g}}}[\boldsymbol{{\mathit{h}}}(\boldsymbol{{\mathit{x}}})\boldsymbol{{\mathrm{\hspace{.167em}-\hspace{.167em}}}{\mathit{h}}}(\boldsymbol{{\mathit{y}}})]\boldsymbol{\hspace{1em}}(\boldsymbol{{\mathit{x}}{\mathrm{\hspace{.167em}{\geq}\hspace{.167em}}}{\mathit{y}}{\mathrm{\hspace{.167em}{\geq}\hspace{.167em}0}}})\boldsymbol{{\mathrm{,}}}\end{equation*}\end{document}

where
h
maps [0, ∞[ onto a subinterval of positive length of [0, ∞[ and is strictly increasing and twice continuously differentiable,
f
and
g
map [0, ∞[ onto [0, ∞[ and are twice differentiable, and either
f
"(0) ≠ 0 or
g
"(0) ≠ 0. The solutions are fully determined under these conditions. When
f
"(0) =
g
"(0) = 0 and
h
" is not identically zero, we determine the solutions under the added assumption of analyticity. It remains an open problem to find the solutions in the latter case under the assumption of only second order differentiability. A more general open problem is to eliminate all differentiability conditions for the second equation.