contrageometric proportion

Just as one converts the proportion equation


defining the harmonic meanMathworldPlanetmath of x and y into the proportion equation


defining their contraharmonic mean (, one also may convert the proportion equation


defining the geometric meanMathworldPlanetmath into a new equation

m-xy-m=ym (1)

defining the contrageometric mean m of x and y.  Thus, the three positive numbers x, m, y satisfying (1) are in contrageometric proportion.  One integer example is 1, 4, 6.

Solving m from (1) one gets the expression

m=x-y+(x-y)2+4y22=:f(x,y). (2)

Suppose now that  0xy.  Using (2) we see that

y2-m2=-(y-x)2+(y-x)(x-y)2+4y22=(y-x)[(y-x)2+4y2-(y-x)]2 0,


xf(x,y)y. (3)

Thus the contrageometric mean of x and y also is at least equal to their arithmetic meanMathworldPlanetmath.  We can also compare m with their quadratic mean by watching the difference

(x2+y22)2-m2=(y-x)(12(y-x)2+4y2-y) 0.

So we have

x+y2f(x,y)x2+y22. (4)

Cf. this result with the comparison of Pythagorean means (; there the brown curve is the graph of  f(x, 1).

It’s clear that the contrageometric mean (2) is not symmetric with respect to the variables x and y, contrary to the other types of means in general.  On the other hand, the contrageometric mean is, as other types of means, a first-degree homogeneous function its arguments:

f(tx,ty)=tf(x,y). (5)


  • 1 Mabrouk K. Faradj: mean do you mean? An exposition on means.  Louisiana State University (2004).
  • 2 Georghe Toader & Silvia Toader: means and the arithmetic-geometric meanDlmfDlmfMathworldPlanetmath.  RGMIA (2010).
Title contrageometric proportion
Canonical name ContrageometricProportion
Date of creation 2013-04-19 7:14:46
Last modified on 2013-04-19 7:14:46
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Definition
Classification msc 26E60
Classification msc 11-00
Classification msc 01A20
Classification msc 01A17
Related topic ContraharmonicProportion
Defines contrageometric mean