Most of us are familiar with the fact that the geometric mean of a finite sequence of numbers is not greater than its arithmetic mean. In this short article, we will generalize this fact and give some applications of our results.
　
        We assume throughout that our functions are monotonic, and that their inverses are defined at relevant points. We also assume that the terms of all sequences (including transformed sequences) are positive.
　
Definition: Given a function f and a sequence of numbers a₁, a₂, ..., a_n, we define the f-mean of the sequence to be f ^{- 1} (), i.e., the f-mean of a finite sequence  a1, a2, ..., an  is  f ^{- 1} of the arithmetic mean of the sequence  f(a₁), f(a₂), ..., f(a_n).

Example 1: If f is the identity function, then the f-mean is simply the arithmetic mean.

Example 2:  If f  is the logarithmic function, then the f-mean is the geometric mean.

        We recall that f is concave up if, given any two points a1, a2, and a number t between 0 and 1, we have  f((1 - t)a₁ + ta₂) <= (1 - t) f(a₁) + t f(a₂).  This is equivalent to the geometric fact that the secant is above the tangent.  By mathematical induction, we can easily prove that if f is concave up, then given any finite sequence  a₁, a₂, ..., a_n, we have
                f() <=  .

If the function is concave down, the corresponding result is obtained by reversing the inequality.

Theorem 1:  If f is increasing and concave up, then the arithmetic mean of a finite sequence is not greater than its f-mean.
Proof:  Let  a1, a2, ..., an  be the finite sequence.
Since f is concave up,
                f() <=  .

As  f is increasing, so is  f ^{- 1}.  Thus
<= f ^{- 1}(),
i.e., the arithmetic mean <= the f-mean.
 

We state three similar theorems.  The proofs are also similar and will be omitted.

Theorem 2:  If  f  is increasing and concave down, then the arithmetic mean of a finite sequence is not less than its f-mean.

Theorem 3:  If  f  is decreasing and concave up, then the arithmetic mean of a finite sequence is not less than its f-mean.

Theorem 4:  If  f  is decreasing and concave down, then the arithmetic mean of a finite sequence is not greater than its f-mean.

Example 3:  Let f be the logarithmic function, which is increasing and concave down.  The f-mean is the same as the geometric mean.  Theorem 2 gives the familiar fact that the arithmetic mean is not less then the geometric mean.

Example 4:  Let f be the reciprocal function, which is decreasing and concave up.  The f-mean is known as the harmonic mean.  Theorem 3 concludes that the arithmetic mean is not less the harmonic mean.

Example 5:  A particle travels between two points A and B.  The speed from A to B is a₁, and the speed from B back to A is a2.  It is easy to verify that the average speed of the round-trip journey is .  By Example 4, we conclude that the average speed is less than or equal to the average of the two speeds a₁ and a₂.

Example 6:  In the financial world, there is a method of investment called 'dollar cost averaging’: investing a fixed amount of money regularly.  It is claimed that a person using ‘dollar cost averaging’ should be better off than one who invests all the amounts at one time.  Suppose a fixed amount A is used to buy shares at prices  a₁, a₂, ..., a_n.  The total number of shares is then .  If one invests the amount  nA  at a time when the share price is the average of  a₁, a₂, ..., a_n, the number of shares which one can purchase is .  By Example 4 again, we conclude that ‘dollar cost averaging’ is at least as good as purchasing at the average share price.
 

        We now present theorems comparing the  f-mean and  g-mean of two functions  f and g.

Theorem 5:  If  f  and  g are increasing,  gf ^{- 1}  is concave up, then the f-mean of a finite sequence is not greater than its g-mean.
Proof:  Since  f  is increasing, so is f ^{- 1}.  Since g and f ^{- 1} are increasing, so is gf ^{- 1}.  By Theorem 1, we have
<= (gf ^-1)^-1().
<= fg ^-1().
f ^-1() <= g^-1().

We now rename each  a_i  by  f(a_i)  in the above inequality and obtain
f ^-1() <= g^-1(),
i.e.,  f-mean <= g-mean.

Similar theorems are summarized in the following table:
 

Example 7:  Let f(x) =  and g(x) = log x.  Then   f ^{- 1}(x) =  and  gf ^{- 1}(x) = log() = -log x.  Then  f  is decreasing,  g is increasing, and  gf ^{- 1} is concave up.  We conclude that  the f-mean <= the g-mean, i.e., the harmonic mean <= the geometric mean.

Example 8:  Let a =  (a₁, a₂, ..., a_n)  be a vector with n components.     For p <= q, a formula relating the p-norm and q-norm is given by  <= n⁽⁾.  This can be easily shown by means of Theorem 5.  Let f(x) = x^p,  g(x) = x^q.  Then  f and g are increasing, and  gf ^{- 1}(x) = x is concave up.  Theorem 5, applied to the sequence  |a₁|, |a₂|, ..., |a_n|,  concludes that  the f-mean <= the g-mean, i.e.,
() <= ().
<= .
<= n^(-).
 

回《數學教育》第五目錄