Characterizations of some rescaling functions (Q1114874)

Let G, H and F be functions from \(R^{+n}\) into \(R^+\), \(n\geq 2\), defined by \(G(x_ 1,...,x_ n)=g^{-1}(\sum^{n}_{i=1}r_ ig(x_ i)),\) \(H(x_ 1,...,x_ n)=h^{-1}(\sum^{n}_{i=1}q_ ih(x_ i))\) and \(F(x_ 1,...,x_ n)=f^{-1}(\sum^{n}_{i=1}p_ if(x_ i)),\) where \(r_ i\), \(q_ i\), \(p_ i\) for \(i=1,...,n\), are positive real numbers and g, h and f are continuous and strictly monotonic functions from \(R^+\) into \(R^+\). The authors solve the following functional equations: G(\(\frac{x_ 1}{G(x_ 1,...,x_ n)},...,\frac{x_ n}{G(x_ 1,...,x_ n)})=1\) for \(x_ 1,...,x_ n\in R^+\) with \(G(x_ 1,...,x_ n)\neq 0\) and \(T(\lambda x_ 1,...,\lambda x_ n)=T(x_ 1,...,x_ n)\) for \(\lambda,x_ 1,...,x_ n\in R^+\), where \[ T(x_ 1,...,x_ n)=[H(x_ 1-F(x_ 1,...,x_ n),...,x_ n-F(x_ 1,...,x_ n))]^{-1} \] \[ (x_ 1-F(x_ 1,...,x_ n),...,x_ n-F(x_ 1,...,x_ n)). \] These equations characterize some rescaling functions arising in data processing.