Equality of two-variable functional means generated by different measures
Two-variable functional means, \(M_{f,g;\mu}\) defined by \[ M_{f,g;\mu}(x,y)= \Biggl({f\over g}\Biggr)^{-1}\left({\int_{[0,1]} f(tx+ (1- t)y)\,d\mu(t)\over \int_{[0,1]} g(tx+ (1- t)y)\,d\mu(t)}\right) \] for real-valued continuous functions \(f\) and \(g\) defined on a real open interval \(I\), \(\mu\) a measure on the Borel sets of \([0,1]\), \(g\) a positive function, and \(f\) strictly \(g\) monotonic, are considered. Note that \(M\) generalizes quasi-arithmetic, Bajraktarević, Lagrangian, and Cauchy means. The main results concern the functional equation \(M_{t,g;\mu}= M_{t,g;\nu}\) for unknown functions \(f\) and \(g\) and with given measures \(\mu\) and \(\nu\). The original functional equation is re-written as \[ \int_{[0,1]}\,\int_{[0,1]} \left|\begin{matrix} f(tx+ (1-t)y)\;& f(sz+ (1- s)y)\\ g(tx+ (1-t)y)\;& g(sx+ (1-s)y)\end{matrix}\right|\,d\mu(t)d\nu(s)= 0.\tag{\(*\)} \] This form of the functional equation allows significant relaxation of the conditions imposed on \(f\), \(g\), \(\mu\), and \(\nu\) in earlier work; thus \((*)\) is studied under minimal assumptions on the two functions and the two measures. Since \((*)\) is trivially satisfied if the two functions are linearly dependent or if the two measures are linearly dependent, the results here concern the linearly independent cases. Necessary and sufficient conditions on the two functions and on finite Borel measures \(\mu\) and \(\nu\) are given for solutions of \((*)\). The particular cases of linearly independent finite Borel measures which are either both symmetric or both antisymmetric are also studied. In addition, the symmetry equatio \(M_{f,g;\mu}(x,y)= M_{f,g;\mu}(y,x)\) is discussed. The paper concludes with three particular applications of \((*)\) obtained from various choices of measures of discrete support. The resulting functions equations (i) \([f(x)- 2f({x+y\over 2})+ f(y)][g(x)+ g(y)]= [g(x)- 2g({x+iy\over 2})+ g(y)][f(x)+ f(y)]\), (ii) \([f(x)- 2f({x+y\over 2})+ f(y)][g(x)- g(y)]=[g(x)- 2g({x+y\over 2})+ g(y)][f(x)- f(y)]\), and (iii) \([f(x)+ f(y)][g(x)- 3g({2x+y\over 3})+ 3g({x+2y\over 3})- g(y)]= (g(x)+ g(y)][f(x)- 3f({2x+y\over 3})+ 3f({x+ 2y\over 3})- f(y)]\). The analytic functions \(f,g: I\to \mathbb{R}\) are determined in each instance.
scientific article; zbMATH DE number 5872782
scientific article