An extension theorem for a Matkowski–Sutô problem
Tom 95 / 2003
Colloquium Mathematicum 95 (2003), 153-161 MSC: 39B22, 39B12, 26A18. DOI: 10.4064/cm95-2-1
Let $I$ be an interval, $0<\lambda <1$ be a fixed constant and $A(x,y)=\lambda x+(1-\lambda ) y,\, x,y \in I,$ be the weighted arithmetic mean on $I$. A pair of strict means $M$ and $N$ is complementary with respect to $A$ if $A(M(x,y),N(x,y))=A(x,y)$ for all $x, y \in I.$ For such a pair we give results on the functional equation $f(M(x,y))=f(N(x,y)).$ The equation is motivated by and applied to the Matkowski–Sutô problem on complementary weighted quasi-arithmetic means $M$ and $N$.