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A paper by Grabisch et al (2011) states that the arithmetic and geometric means are Lagrangian means whereas the harmonic mean is not Lagrangian.

What does this mean? What properties does a Lagrangian mean have and what properties do non-Lagrangian means have?

References

  • Michel Grabisch, Jean-Luc Marichal, Radko Mesiar, Endre Pap. Aggregation functions: Means. Information Sciences, Elsevier, 2011, 181 (1), pp.1-22. <10.1016/j.ins.2010.08.043>.
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    $\begingroup$ Your title and question do not match. Please fix. $\endgroup$ – kjetil b halvorsen Feb 26 '18 at 5:57
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    $\begingroup$ In its current edited form this is quite unclear. Some of the detail you deleted might be helpful for people to know just what your problem is. $\endgroup$ – mdewey Feb 26 '18 at 16:52
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Skimming through the document, we see that Definition 27 defines the Lagrangian mean associated with a given continuous and strictly monotonic function $f$ on the unit interval.

At other places, the geometric and arithmetic mean are identified as Lagrangian. That is, there are functions $f_g$ and $f_a$ such that the geometric and arithmetic mean are exactly the Lagrangian means associated to $f_g$ and $f_a$ in the sense of Definition 27.

Conversely, it is claimed that the harmonic mean is not Lagrangian. This means that there is no such function $f_h$ such that the harmonic mean is the Lagrangian mean associated with $f_h$.

Now it would probably be a good exercise for you to carefully prove these three statements. Since they are not explicitly proven in the paper, I assume they are not very hard to show.

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