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Functional means and harmonic functional means

Published online by Cambridge University Press:  17 April 2009

Soon-Yeong Chung
Soon-Yeong Chung
Affiliation:
Department of MathematicsSogang UniversitySeoul 121–742Korea e-mail: sychung@ccs.sogang.ac.kr
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Abstract

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For a continuous function f(t) on (0, ∞) which is strictly monotone and a probability measure μ on [0, 1] we introduce the functional mean and the harmonic functional mean of x > 0 and y > 0 with respect to μ by

which gives a unified approach to various famous means.

Moreover, functional means and harmonic means in n variables are also given and applied to get many interesting properties, such as

where

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 57 , Issue 2 , April 1998 , pp. 207 - 220
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Borwein, J.M. and Borwein, P.B., ‘Inequalities for compound mean iterations with logarithmic asymptotes’, J. Math. Anal. Appl. 177 (1993), 572–582.CrossRefGoogle Scholar
[2]Burk, F., ‘Geometric, logarithmic, and arithmetic mean inequalities’, Amer. Math. Monthly 94 (1987), 523–528.Google Scholar
[3]Carlson, B.C., ‘The logarithmic mean’, Amer. Math. Monthly 79 (1972), 615–618.Google Scholar
[4]Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities (Cambridge Univ. Press, Cambridge, 1934).Google Scholar
[5]Pittenger, A. O., ‘The logarithmic mean in n variables’, Amer. Math. Monthly 92 (1985), 99–104.CrossRefGoogle Scholar
[6]Sandor, J., ‘On the identric and logarithmic means’, Aequationes Math. 40 (1990), 261–270.CrossRefGoogle Scholar
[7]Sandor, J., ‘A note on some inequalities for means’, Arch. Math. 56 (1991), 471–473.CrossRefGoogle Scholar
[8]Sandor, J., ‘On certain inequalities for means’, J. Math. Anal. Appl. 189 (1995), 602–606.CrossRefGoogle Scholar
[9]Sandor, J., ‘On certain inequalities for means II’, J. Math. Anal. Appl. 189 (1996), 629–635.Google Scholar
[10]Vamanamurthy, M. K. and Vuorinen, M., ‘Inequalities for means’, J. Math. Anal. Appl. 183 (1994), 155–166.CrossRefGoogle Scholar