# What is the name of this formula and how to derive the limit?

There is a formula :

$M = (M_1 (1-f_2)^a + M_2 f_2^a)^\frac{1}{a}$

when $a$ goes to zero, the above formula becomes

$M = M_1 ^{1-f_2} M_2 ^{f_2}$

The two phases can be extended to N phases:

$M = (M_1 f_1^a + M_2 f_2^a+M_3 f_3^a+...+M_n f_n^a)^\frac{1}{a}$

When $a$ goes to zero, the above formula becomes

$M=M_1^{f_1}M_2^{f_2}M_3^{f_3}...M_n^{f_n}$

where $f_1+f_2+f_3+...f_n=1$

What is the name of this formula and how to derive the limit when $a$ goes to zero? (I forgot the name of this special formula, I came across it on Wiki several years ago) Thanks a lot!

• If you look for a generic name, it is "means" (for example the first formula with $f_1=\dfrac12$ and $a=2$ gives the "quadratic mean" ; the second formula, also for $f_1=\dfrac12$, is the "geometric mean"). But why are you interested by the fact to give a name to this limit process between means ? Is it because you have seen that before ? Feb 13, 2017 at 22:27
• Thanks a lot for your attention. I remember it is named after a mathematician. Feb 13, 2017 at 22:34
• I think taht the name you look for is Hölder. See (en.wikipedia.org/wiki/Generalized_mean) Feb 13, 2017 at 22:41
• Thanks a lot, that is an important information for me! Feb 13, 2017 at 22:46