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The Quasi-Arithmetic Means

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Means and Their Inequalities

Part of the book series: Mathematics and Its Applications ((MAIA,volume 31))

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Abstract

The power means [r]n (a;w), rεR, defined in the previous chapter can be looked at in the following way; for each rεR define a function φ as follows: Φ(x) = xr, r ≠ 0, Φ(x) = log x, r = 0, then

$$M_n^{[r]}(\underline a ;\underline w ) = {\phi ^{ - 1}}\quad (\frac{1}{{{w_n}}}\sum\limits_{i = 1}^n {{w_i}\;\phi ({a_i})} ).$$
((1))

This suggests the following definitions.

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© 1988 Springer Science+Business Media Dordrecht

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Bullen, P.S., Mitrinović, D.S., Vasić, P.M. (1988). The Quasi-Arithmetic Means. In: Means and Their Inequalities. Mathematics and Its Applications (East European Series) , vol 31. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2226-1_4

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  • DOI: https://doi.org/10.1007/978-94-017-2226-1_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-017-2228-5

  • Online ISBN: 978-94-017-2226-1

  • eBook Packages: Springer Book Archive

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