On homogeneous Lagrange means |
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Authors: | Janusz Matkowski |
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Institution: | 1. Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul Prof. Z. Szafrana 5a, 65-516?, Zielona Góra, Poland
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Abstract: | Let \(f\) be a real differentiable function in an open interval \(I\) with one-to-one derivative. We observe that if the Lagrange mean \(L^{f]}\) of a generator \(f\) is conditionally positively homogeneous, then \(f\) must be of the class \(C^{\infty }\) and the function $$\begin{aligned} g(x):=xf^{\prime }\left( x\right) -f\left( x\right) ,\quad \quad x\in I, \end{aligned}$$ is also a generator of \(L^{f]}\) i.e. that \(L^{g]}=L^{f]}.\) We show that this fact and a result on equality of two Lagrange means allow easily to determine all positively homogeneous Lagrange means. |
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