If $u$ and $v$ are positive numbers and $u \leqq v$ , then their Pythagorean means, viz. the harmonic mean $h(u,v)$ , the geometric mean $g(u,v)$ , the arithmetic mean $a(u,v)$ and the contraharmonic mean $c(u,v)$ , obey the order

(1)

(2)

The below diagram plots the means $h(x,1)$ in black, $g(x,1)$ in blue, $a(x,1)$ in cyan and $c(x,1)$ in green for $0 \leqq x \leqq 1$ .

Note, that the linear graph of the arithmetic mean is the common tangent all those curves in the point $(1,1)$ , since here the derivatives of all functions have the value $\frac{1}{2}$ . The same concerns the yellow graph of the Heronian mean of $x$ and $1$ , similarly the red graph of the quadratic mean.

Bibliography

1

HORST HISCHER: ``Viertausend Jahre Mittelwertbildung''. -- mathematica didactica 25 (2002). See also this.