Article

Aequationes mathematicae

, Volume 81, Issue 1, pp 31-53

First online:

Equality of two-variable functional means generated by different measures

  • László LosoncziAffiliated withFaculty of Economics and Business Administration, Debrecen University Email author 
  • , Zsolt PálesAffiliated withInstitute of Mathematics, Debrecen University

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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 M f,g;μ = M f,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