Prove connection between power mean and arithmetic mean of a function

Let $f$ be a continuous, strictly monotonic function on $\mathbb{R}_{>0}$ with inverse $g$ , and let $a_1 < a_2 < \cdots < a_n$ be given positive real numbers. Then define,

$M_f = g \left( \frac{1}{n} \sum_{i=1}^n f(a_i) \right).$

This $M_f$ is called the mean of $a_1, \ldots, a_n$ with respect to $f$ . (When $f(x) = x^p$ for $p \neq 0$ , this coincides with the $p$ th power mean from this exercise).

Show that

$f(M_f) = \frac{1}{n} \sum_{i=1}^n f(a_i).$

Proof. Since $g$ is the inverse of $f$ we know $f(g(x)) = x$ for all $x$ in the range of $f$ , i.e., for all $x$ such that there is some $c \in \mathbb{R}_{>0}$ such that $f(c) = x$ .

By the definition of $M_f$ then, we have that

$f(M_f) = f \left( g \left( \frac{1}{n} \sum_{i=1}^n f(a_i) \right) \right).$

So, if $\frac{1}{n} \sum_{i=1}^n f(a_i)$ is in the domain of $g$ then we are done. Since $g$ is the inverse of $f$ it’s domain is equal to the range of $f$ . We show that this value is in the range of $f$ using the intermediate value theorem.

Without loss of generality, assume $f$ is strictly increasing (the alternative assumption, that $f$ is strictly decreasing will produce an almost identical argument). Then, since $a_1 < a_2 < \cdots < a_n$ are all positive real numbers we have $f(a_1) < f(a_2) < \cdots < f(a_n)$ . (Here if we’d assumed that $f$ was strictly decreasing the roles inequalities would be reversed.) Then we have,

$f(a_1) = \frac{1}{n} \sum_{i=1}^n f(a_1) < \frac{1}{n} \sum_{i=1}^n f(a_i) < \frac{1}{n} \sum_{i=1}^n f(a_n) = f(a_n).$

Hence, by the intermediate value theorem, since

$\frac{1}{n} \sum_{i=1}^n f(a_i) \in [f(a_1), f(a_n)]$

there must be some $c \in \mathbb{R}_{>0}$ such that

$f(c) = \frac{1}{n} \sum_{i=1}^n f(a_i).$

Thus, $\frac{1}{n} \sum_{i=1}^n f(a_i)$ is in the domain of $g$ , so

$f(M_f) = f \left( g \left( \frac{1}{n} \sum_{i=1}^n f(a_i) \right) \right) = \frac{1}{n} \sum_{i=1}^n f(a_i). \qquad \blacksquare$

Stumbling Robot

Prove connection between power mean and arithmetic mean of a function

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