# zbMATH — the first resource for mathematics

Quadratic means. (English) Zbl 1041.26015
The paper considers expressions of the form $${\mathfrak M} = f_1 \mp\sqrt{f_2}$$ where $$f_1, f_2$$, is a symmetric form of degree $$1, 2,$$ in $$n$$ variables. The case where $$f_1=0$$ has been considered by the second author [Math. Inequal. Appl. 6, 581–593 (2003; Zbl 1047.26015)]. Using suitable bases for the two spaces of forms these expressions can be given in terms of two real numbers. The authors find necessary and sufficient conditions on these numbers for $${\mathfrak M}$$ to be a mean. The 2 basis elements for the space of symmetric forms of degree $$2$$ are related in a simple way to the mean and standard deviation and the simple internality of the means leads to an improvement of the Brunk inequalities. Comparability of two means of this type is studied as well as the behaviour of the means under equal increments of the variables.

##### MSC:
 26E60 Means 26D15 Inequalities for sums, series and integrals
##### Keywords:
quadratic means; Brunk inequalities; equal increments
Full Text:
##### References:
 [1] Aczél, J.; Dhombers, J., Functional equations in several variables, (1989), Cambridge Univ. Press [2] Aczél, J.; Losonczi, S.; Páles, Zs., The behaviour of comprehensive classes of means under equal increments of their variables, Numer. math., 80, 459-461, (1987) · Zbl 0638.26014 [3] Aczél, J.; Páles, Zs., The behaviour of means under equal increments of their variables, Amer. math. monthly, 95, 856-860, (1988) · Zbl 0671.26008 [4] Borwein, J.M.; Borwein, P.B., Pi and the AGM—A study in analytic number theory and computational complexity, (1986), Wiley New York · Zbl 0613.65010 [5] Bullen, P.S., Averages still on the move, Math. mag., 63, 250-255, (1990) · Zbl 0736.26009 [6] Bullen, P.S.; Mitrinović, D.S.; Vasić, P.M., Means and their inequalities, (1988), Reidel Dordrecht · Zbl 0687.26005 [7] Farnsworth, D.; Orr, R., Gini means, Amer. math. monthly, 93, 603-607, (1986) · Zbl 0612.26015 [8] Farnsworth, D.; Orr, R., Transformation of power means and a new class of means, J. math. anal. appl., 129, 394-400, (1988) · Zbl 0638.26015 [9] Hajja, M., Radical and rational means of degree two, Math. inequal. appl., (2003), in press · Zbl 1047.26015 [10] Hardy, G.H.; Littlewood, J.E.; Pólya, G., Inequalities, (1952), Cambridge Univ. Press · Zbl 0047.05302 [11] Hoehn, L.; Niven, I., Averages on the move, Math. mag., 58, 151-156, (1985) · Zbl 0601.26011 [12] Jensen, S.T.; Styan, G.P.H., Some comments and a bibliography on the laguerre – samuelson inequality with extensions and applications in statistics and matrix theory, () · Zbl 0980.15016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.