Abstract
In this paper we give necessary and sufficient conditions for the inequality
where 0 < α < β < ∞ are fixed values and M: ℝ+ × ℝ+ → ℝ+ and N: ℝ+ × ℝ+ → ℝ+ belong to one of the following classes of means:
The following two inequalities are simple necessary conditions for (*) to hold:
In the class of power means, already the first inequality is a sufficient condition. The aim of this paper is to show that these two inequalities together are sufficient conditions for the comparison of the above D and S means as well.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
J. L. Brenner, A unified treatment and extension of some means of classical analysis I. Comparison theorems. J. Combin. I.form. System Sci. 3 (1978), 175–199.
F. Burk, By all means. Amer. Math. Monthly 92 (1985), 50.
B. C. Carlson, The logarithmic mean. Amer. Math. Monthly 79 (1972), 615–618.
E. L. Dodd, Some generalizations of the logarithmic mean and of similar means of two variates which become indeterminate when the two variates are equal. Ann. Math. Statist. 12 (1971), 422–428.
C. Gini, Di una formula comprensiva delle medie. Metron, 13 (1938), 3–22.
E. Leach and M. Sholander, Extended mean values. Amer. Math. Monthly 85 (1978), 84–90.
E. Leach and M. Sholander, Extended mean values II. J. Math. Anal. Appl. 92 (1983), 207–223.
T. P. Lin, The power mean and the logarithmic mean. Amer. Math. Monthly 81 (1974), 879–883.
Z. Pâles, Inequalities for differences of powers. J. Math. Anal. Appl., 131 (1988), 271–281.
Z. Pâles, Inequalities for sums of powers. J. Math. Anal. Appl. 131 (1988), 265–270.
Zs. Pâles, On comparison of homogeneous means. Annales Univ. Sci. 32 (1989), 261–266.
A. O. Pittinger, Inequalities between arithmetic and logarithmic means. Univ. Beograd Publ. Elektrotechn. Fak. Ser. Mat. Fiz. 680 (1980), 15–18.
K. B. Stolarsky, Generalization of the logarithmic mean. Math. Mag. 48 (1975), 87–92.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1992 Springer Basel AG
About this chapter
Cite this chapter
Páles, Z. (1992). Comparison of two variable homogeneous means. In: Walter, W. (eds) General Inequalities 6. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 103. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7565-3_6
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7565-3_6
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7567-7
Online ISBN: 978-3-0348-7565-3
eBook Packages: Springer Book Archive