Invariance of the arithmetic mean with respect to special mean-type mappings

Summary.

For a continuous and strictly monotonic function f defined on an interval I of positive reals, the function $M_{f} {\left( {x,y} \right)} = f^{{ - 1}} {\left( {\frac{{xf(x) + yf(y)}} {{x + y}}} \right)},x,y\, \in {I}$ is a mean. Assuming that at least one of f and g is four times continuously differentiable, we prove that if the arithmetic mean A is (M _f ,M _g )-invariant, i.e. if A ○ (M _f ,M _g ) = A on I, then

$$ f(x) = C_{1} x^{{ - 1}} + B_{1} \,{\text{and}}\,{\text{g(x) = C}}_{{\text{2}}} x^{{ - 1}} + B_{2} , $$

for some $C_{{\text{1}}} ,C_{2} ,B_{1} ,B_{{\text{2}}} , \in \mathbb{R},C_{1}C_{2} \ne 0$. It remains an open question, if the smoothness conditions on the functions f and g can be relaxed.