Aequationes mathematicae

, Volume 81, Issue 1–2, pp 31–53

Equality of two-variable functional means generated by different measures

Article

Abstract

We consider two-variable functional means of the form
$$M_{f,g;\mu}(x,y) := \left(\frac{f}{g}\right)^{-1}\left(\frac{\int\nolimits_{[0,1]} f(tx+(1-t)y)\,d\mu(t)}{\int\nolimits_{[0,1]}g(tx+(1-t)y)\,d\mu(t)}\right),$$
where f, g are continuous functions on a real interval such that g is positive, f/g is strictly monotonic and μ is a measure over the Borel sets of [0,1]. The main results concern the functional equation Mf,g;μ = Mf,g;ν for the unknown functions f, g, where μ and ν are given measures. Depending on the symmetry properties of the measures, various necessary conditions and sufficient conditions are established.

Mathematics Subject Classification (2000)

Primary 39B22 

Keywords

Functional means functional equation equality problem of means symmetric means 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Faculty of Economics and Business AdministrationDebrecen UniversityDebrecenHungary
  2. 2.Institute of MathematicsDebrecen UniversityDebrecenHungary

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