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On a functional equation appearing on the margins of a mean invariance problem. (English) Zbl 1462.26035

Summary: Given a continuous strictly monotonic real-valued function \(\alpha \), defined on an interval \(I\), and a function \(\omega \) : \(I \rightarrow \) (0, +\( \infty )\) we denote by \(B^\alpha_\omega\) the Bajraktarević mean generated by \(\alpha\) and weighted by \(\omega\): \[ B_\omega^\alpha(x,y) = \alpha^{- 1}\left(\frac{\omega (x)}{\omega (x) + \omega (y)} \alpha (x) + \frac{\omega(y)} {\omega (x) + \omega (y)} \alpha (y) \right),\quad x,y \in I. \] We find a necessary integral formula for all possible three times differentiable solutions \((\varphi,\psi )\) of the functional equation \[ r(x)B_s^\varphi (x,y) + r(y) B_t^{\psi} (x,y) = r(x) x + r (y)y, \] where \(r, s, t: I \rightarrow (0, + \infty)\) are three times differentiable functions and the first derivatives of \(\varphi, \psi\) and \(r\) do not vanish. However, we show that not every pair \(( \varphi, \psi )\) given by the found formula actually satisfies the above equation.

MSC:

26E60 Means
39B22 Functional equations for real functions
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References:

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