Hostname: page-component-7479d7b7d-t6hkb Total loading time: 0 Render date: 2024-07-12T23:49:38.607Z Has data issue: false hasContentIssue false
  • English
  • Français

On Weighted Geometric Means

Published online by Cambridge University Press:  20 November 2018

Horst Alzer
Horst Alzer*
Affiliation:
Horst Alzer, Morsbacher Str. 10, 5220 Waldbröl, Federal Republic of Germany
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The aim of this paper is two-fold: First we prove the Radotype inequality Here denote the weighted geometric means of

with where the pi are positive weights. Thereafter we investigate under which conditions the sequence

is convergent as n → ∞

Keywords

26D15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 2 , 01 June 1989 , pp. 199 - 206
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Alzer, H., liber Mittelwerte, die zwischen dem geometrischen und dem logarithmischen Mittel zweier Zahlen liegen, Anz. Ôsterr. Akad. Wiss., math.-naturw. Kl. 123 (1986), 5–9.Google Scholar
2. Alzer, H., Two inequalities for means, C.R. Math. Rep. Acad. Sci. Canada 9 (1986), 11-16.Google Scholar
3. Alzer, H., On an inequality of Ky Fan, J. Math. Anal. Appl. 137 (1989), 168-172. appear).Google Scholar
4. Alzer, H., On Stolarsky's mean value family, Int. J. Math. Educ. Sci. Technol. (to appear).Google Scholar
5. Beckenbach, E. F., and Bellman, R., Inequalities, Springer, Berlin, 1983.Google Scholar
6. Bullen, P. S., Rado's inequality, Aequat. Math. 5 (1971), 149-156.Google Scholar
7. Bullen, P. S., An inequality ofN. Levinson, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 412-460 (1973), 109–112.Google Scholar
8. Hardy, G. H., Littlewood, J. E., and G. Pôlya, Inequalities, Cambridge Univ. Press, Cambridge, 1952.Google Scholar
9. Kralik, D., Ûber einige Verallgemeinerungsmoglichkeiten des logarithmischen Mittels zweier Zahlen, Per. Polytechn. Chemical Eng. 16 (1972), 373–379.Google Scholar
10. Leach, E. B., and Sholander, M. C., Extended mean values, Amer. Math. Monthly 85 (1978), 84-90.Google Scholar
11. Leach, E. B., Extended mean values II, J. Math. Anal. Appl. 92 (1983), 207-223.Google Scholar
12. Leach, E. B., Multi-variable extended mean values, J. Math. Anal. Appl. 104 (1984), 390–407.Google Scholar
13. Levinson, N., Generalization of an inequality of Ky Fan, J. Math. Anal. Appl. 8 (1964), 133-134.Google Scholar
14. McAdams, W. H., Heat Transmission, McGraw-Hill, New York, 1954.Google Scholar
15. Mitrinovic, D. S., Analytic Inequalities, Springer, Berlin, 1970.Google Scholar
16. Pittenger, A. O., The logarithmic mean in n variables, Amer. Math. Monthly 92 (1985), 99-104.Google Scholar
17. P, G.ôlya, and Szego, G., Isoperimetric Inequalities in mathematical physics, Princeton, 1951.Google Scholar
18. Popoviciu, T., Asupa una inegalitâti intre medii, Acad. Romine, R. P. Fil. Cluj. Stud. Cere. Mat. 11 (1960), 343–355.Google Scholar
19. Popoviciu, T., Sur une inégalité de N. Levinson, Mathematica (Cluj) 6 (1964), 301-306.Google Scholar
20. Stolarsky, K. B., Generalizations of the logarithmic mean, Math. Mag. 48 (1975), 87-92.Google Scholar
21. Vasic, P. M., and Mijalkovic, Z., On an index set function connected with Jensen inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 544-576(1976), 110–112.Google Scholar
22. Vasic, P. M., and Pecaric, J. E., On the Jensen inequality, Univ. Beograd. Publ.Elektrotehn. Fak. Ser. Mat. Fiz. 634-677 (1979), 50–54.Google Scholar
23. Wang, C.-L., On a Ky Fan inequality of the complementary A-G type and its variants, J. Math. Anal. Appl. 73 (1980), 501–505.Google Scholar
24. Wang, C.-L., Functional equation approach to inequalities II, J. Math. Anal. Appl. 78 (1980), 522-530.Google Scholar
25. Wang, C.-L., Inequalities of the Rado-Popoviciu type for functions and their applications, J. Math. Anal. Appl. 100 (1984), 436–446.Google Scholar