aequationes mathematicae

, Volume 71, Issue 1–2, pp 70–85

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

Research Paper


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 (Mf ,Mg)-invariant, i.e. if A ○ (Mf ,Mg) = 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.

Mathematics Subject Classification (2000).

Primary 26E60 39B22 


Mean equality of means invariant mean functional equation differential equation 


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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of GdańskGdańskPoland
  2. 2.Institute of MathematicsUniversity of Zielona GóraZielona GóraPoland
  3. 3.Institute of Mathematics, of the Silesian UniversityKatowicePoland

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