# 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:

• 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.
• Probably it will be better to use the standard convention for $M_p$ to prevent confusion, and define $f(X;p)$ in terms of $M_p$ ? – lisyarus Feb 25 at 14:50