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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 pth 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*}

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