Interpolation between "arithmetic mean" and "geometric mean" Define a function $f(X; p)$, which measures a statistic on a set of points $X = {x_1, ..., x_n}$ (and $x_i \in (0, 1)$), with respect to a parameter $p$. I am looking for ways to define the function $f(.)$, such that on one extreme it becomes arithmetic mean, and on the other extreme it becomes a geometric mean:


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*if $p=0$ then $ f(X; p=0) =  \frac{1}{n}\sum_i x_i $

*if $p=1$ then $ f(X; p=0) = (\prod_i x_i)^{1/n} $
Note: I am not looking for an "interpolation"; but rather a function that could be deformed into either "means" (similar to how an $L_p$ norm can be multiple different operations, depending on the value of $p$). 

$$f(X;p) = M_{1-p}(x_1, \ldots, x_n)=\left( \frac1n \sum_{i=1}^n x_i^{1-p}\right)^\frac1{1-p}$$
When $p=0$, we clearly have the AM. 
Also, $M_p$ also satisfies the condition that $\lim\limits_{p \to 1}M_{1-p}(x_1, \ldots, x_n)=\lim\limits_{p \to 0}M_p(x_1, \ldots, x_n)$ to be equal to the geometric mean. 
This is known as the generalized mean.
