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Lagrangian means. (English) Zbl 0903.39006
The authors call \(M(x,y)\) a Lagrangian mean on \(I^{2}\) if there exists a continuous strictly monotonic function \(f\) such that \(f(M(x,y))=\int_{x}^{y}f(t) dt/(y-x)\) if \(x\neq y\), while \(M(x,x)=x;\) throughout \(x,y\in I\). R. Bojanić gave necessary and sufficient conditions for \(M\) to be a Lagrangian mean under the supposition that \(M\) and \(f\) are twice differentiable. The authors offer a necessary and sufficient condition without such differentiability conditions, and a connection between these means and quasiarithmetic means \((2g(Q(x,y)) = g(x)+g(y))\).

MSC:
39B22 Functional equations for real functions
26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
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