Abstract
In the 1960s Cargo and Shisha introduced a metric in a family of quasi-arithmetic means defined on a common interval as the maximal possible difference between these means taken over all admissible vectors with corresponding weights. During the years 2013–2016 we proved that, having two quasi-arithmetic means, we can majorize the distance between them in terms of the Arrow–Pratt index. In this paper we are going to prove that this operator can also be used to establish certain lower bounds of this distance.
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References
Bullen, P.S.: Handbook of means and their inequalities, mathematics and its applications. Kluwer Academic Publishers Group, Dordrecht (2003)
Cargo, G.T., Shisha, O.: Bounds on ratios of means. J. Res. Natl. Bur. Stand B Math. Math. Phys. 66B(4), 169–170 (1962)
Cargo, G.T., Shisha, O.: A metric space connected with generalized means. J. Approx. Theory 2(2), 207–222 (1969)
de Finetti, B.: Sul concetto di media. Giornale dell’ Instituto, Italiano degli Attuarii 2, 369–396 (1931)
Knopp, K.: Über Reihen mit positiven Gliedern. J. Lond. Math. Soc. 3, 205–211 (1928)
Kolmogorov, A.N.: Sur la notion de la moyenne. Rend. Accad. dei Lincei 6(12), 388–391 (1930)
Mikusiński, J.G.: Sur les moyennes de la forme \(\psi ^{-1}[\sum q\psi (x)]\). Stud. Math. 10(1), 90–96 (1948)
Nagumo, M.: Über eine Klasse der Mittelwerte. Jpn. J. Math. 7, 71–79 (1930)
Pasteczka, P.: When is a family of generalized means a scale? Real Anal. Exchange, 38(1):193–209 (2012/13)
Pasteczka, P.: A new estimate of the difference among quasi-arithmetic means. Math. Inequal. Appl. 18(4), 1321–1327 (2015)
Pasteczka, P.: On negative results concerning Hardy means. Acta Math. Hungar. 146(1), 98–106 (2015)
Pasteczka, P.: Scales of quasi-arithmetic means determined by an invariance property. J. Differ. Equ. Appl. 21(8), 742–755 (2015)
Pasteczka, P.: Iterated quasi-arithmetic mean type mappings. Colloq. Math. 144(2), 215–228 (2016)
Páles, Z.: On the convergence of means. J. Math. Anal. Appl. 156(1), 52–60 (1991)
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Pasteczka, P. Lower estimation of the difference between quasi-arithmetic means. Aequat. Math. 92, 7–24 (2018). https://doi.org/10.1007/s00010-017-0513-8
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DOI: https://doi.org/10.1007/s00010-017-0513-8