06/14/2011, 09:17 AM
(06/14/2011, 04:22 AM)Cherrina_Pixie Wrote: I thought about this for some time and considered interpolation between arithmetic mean and geometric mean, coming to a rather curious result. The 'mean' function with \( \sigma = -1 \) fails to satisfy a property of means: \( mean(c*r_1,c*r_2,c*r_3, ..., c*r_n) = c*mean(r_1,r_2,r_3, ..., r_n) \)
Define \( M_f^\sigma(r_1,r_2,r_3, ..., r_n) = \exp_f^{\circ \sigma}\left(\frac{\exp_f^{\circ -\sigma}(r_1) + \exp_f^{\circ -\sigma}(r_2) + \exp_f^{\circ -\sigma}(r_3) + ... + \exp_f^{\circ -\sigma}(r_n)}{n}\right),\ \sigma \le 1 \) This yields the arithmetic mean for \( \sigma = 0 \) and the geometric mean for \( \sigma = 1 \).
I just want to add the observation that:
\( M^1 \) and \( M^2 \) satisfy the modified property
\( M(r_1^c,\dots,r_n^c)=M(r_1,\dots,r_n)^c \).
