, Volume 81, Issue 1-2, pp 31-53
Date: 11 Jan 2011

Equality of two-variable functional means generated by different measures

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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.

This research has been supported by the Hungarian Scientific Research Fund (OTKA) Grant NK81402 and by the TÁMOP 4.2.1./B-09/1/KONV-2010-0007 project implemented through the New Hungary Development Plan co-financed by the European Social Fund, and the European Regional Development Fund.
Dedicated to the 85th birthday of Professor János Aczél