On the characterization of quasiarithmetic means with weight function (Q1822285)

The classical symmetric quasiarithmetic means \(f^{-1}(\sum^{n}f(x_ k)/n)\) have been characterized by \textit{A. N. Kolmogorov} [Accad. Naz. Lincei, Mem. Cl. Sci. Fis. Mat. Nat., II. Ser. 12, 388-391 (1930)] and \textit{M. Nagumo} [Jap. J. Math. 7, 235-253 (1930, not 1931 as asserted on p. 172 of the present paper; the reference on p. 194 is correct)]. For a characterization of weighted quasiarithmetic means \(f^{-1}(\sum^{n}p_ kf(x_ k)/\sum^{n}p_ k)\) \((p_ 1,...,p_ n>0)\) (of a fixed number of variables) see the reviewer [Bull. Am. Math. Soc. 54, 392-400 (1948; Zbl 0030.02702)]. In this paper the author offers the following nice characterization (necessary and sufficient conditions) for quasiarithmetic means with weight function (really M is a sequence of 1- ,2-,...,n-,... place functions): \[ M(x_ 1,...,x_ n)=f^{- 1}(\sum^{n}p(x_ k)f(x_ k)/\sum^{n}p(x_ k)) \] (f continuous strictly monotonic, p positive on a proper closed interval I, \(x_ 1,...,x_ n\in I):\) M(x)\(\equiv x\); \(M(x_ 1,...,x_ n)\) is symmetric; for any \(x<u<v<y\) in I there exist m, n, such that \(u<M(x,...,x,y,...,y)<v\) (n-times x, m-times y); for all \(x_ 1,...,x_ n,y_ 1,...,y_ m\in I\) \[ \lim_{j\to \infty}M(x_ 1,...,x_ 1,...,x_ n,...,x_ n,y_ 1,...,y_ m)=M(x_ 1,...,x_ n)\quad (j- times\quad x_ i); \] M(x\({}_ 1,...,x_ n,u_ 1,...,u_ i)\leq M(x_ 1,...,x_ n,v_ 1,...,v_ j)\) and \(M(y_ 1,...,y_ m,u_ 1,...,u_ i)\leq M(y_ 1,...,y_ m,v_ 1,...,v_ j)\) imply \(M(x_ 1,...,x_ n,u_ 1,...,u_ i,y_ 1,...,y_ m,u_ 1,...,u_ i)\leq M(x_ 1,...,x_ n,v_ 1,...,v_ j,\) y\({}_ 1,...,y_ m,v_ 1,...,v_ j)\). Finally all such means are determined which are also linearly homogeneous. \{In line 20 of page 177, \(mk=m'k\) should stand rather than \(mk'=m'k'.\}\)