An application of a uniqueness theorem to a functional equation arising from measuring utility (Q1276370)

\textit{R. D. Luce} [J. Math. Psychol. 40, 297-317 (1996; Zbl 0887.92041)] reduced to three functional equations the question when four distinct ways of measuring utility are the same. Two of these equations have been solved by \textit{J. Aczél, Gy. Maksa} and \textit{R. D. Luce} [J. Math. Anal. Appl. 204, 451-471 (1996; Zbl 0870.90019)] under natural conditions arising from the applied problem. In the same paper the third equation, \(H(x,y)z=H[xz,yP(x,z)]\) with \(H(x,y)=G^{-1}[G(x)G(y)]\) \((x,y\in[0,1[\), \(z\in[0,1])\), where \(P\) maps \([0,1[\times[0,1]\) into \([0,1],\) has been solved for \(P\) under the conditions, also natural for the applications, that \(G:[0,1[\to]0,1]\) is strictly decreasing and onto (thus continuous). The function \(G\) itself, however, could be determined in that paper only under the condition that it has nonzero derivative everywhere. Several attempts were made (mainly at meetings, for instance by \textit{N. Brilouët-Belluot} [Aequationes Math. 53, 162-205 (1997), in particular pp. 185-187 and 55, 281-318 (1998), in partic. p. 310, also 56, 284-318 (1998), in partic. p. 299] to weaken the latter condition. The present paper constitutes the first significant step in that direction: the condition of nonzero differentiability of \(G\) is replaced by that of partial differentiability of \(P\) in its second variable. The author skilfully uses his results in Proc. Am. Math. Soc. 39, 525-529 (1973; Zbl 0272.39009) and in Ann. Polon. Math. 27, 329-336 (1973; Zbl 0251.39003)] to get this. [Note: In a paper accepted for publication in the J. Math. Anal. Appl., \textit{Zs. Páles, Gy. Maksa} and the reviewer have solved the problem completely, eliminating any conditions other than the strictly decreasing surjectivity of \(G:[0,1[\to]0,1]\). The general solution \(G(x)=(1-x^{b})^{a}\) for all \(x\in[0,1[\), where \(a\) and \(b\) are arbitrary positive constants, is the same as that found under differentiability conditions].