Prove that different functions may have the same average

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 if $h(x) = af(x) + b$ with $a \neq 0$ , then $M_h = M_f$ .

Proof. Let $h(x) = af(x) + b$ with $a \neq 0$ . Then, $h$ has an inverse since it is strictly monotonic (since it is the composition of $f$ and the linear function $ax+b$ , both of which are strictly monotonic for $a \neq 0$ ). Its inverse is given by

$h^{-1}(x) = f^{-1} \left( \frac{x-b}{a} \right).$

So,

$\begin{align*} M_h &= h^{-1} \left( \frac{1}{n} \sum_{i=1}^n h(a_i) \right) \\ &= h^{-1} \left( \frac{1}{n} \sum_{i=1}^n (af(a_i) + b) \right) \\ &= h^{-1} \left( \frac{a}{n} \left(\sum_{i=1}^n f(a_i)\right) + b \right)\\ &= f^{-1} \left(\frac{\frac{a}{n} \left(\sum_{i=1}^n f(a_i)\right) + b - b}{a} \right)\\ &= g \left( \frac{1}{n} \sum_{i=1}^n f(a_i) \right) \\ &= M_f. \qquad \blacksquare \end{align*}$

Stumbling Robot

Prove that different functions may have the same average

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