📂Lemmas

Harmonic Mean

Definition

For a positive number $a,b > 0$, the following is called the Harmonic Mean. $$H (a,b) := 2 \left( {{ 1 } \over { a }} + {{ 1 } \over { b }} \right)^{-1} = {{ 2 ab } \over { a + b }}$$ The generalized form with $n$ terms $x_{1} , \cdots , x_{n}$ is as follows. $$H \left( x_{1} , \cdots , x_{n} \right) := n \left( \sum_{k=1}^{n} x_{k}^{-1} \right)^{-1}$$

Theorem

Arithmetic-Geometric-Harmonic Mean Inequality

$$\frac { {x}_{1}+{x}_{2}+\cdots+{x}_{n} }{ n }\ge \sqrt [ n ]{ {x}_{1}{x}_{2}\cdots{x}_{n} }\ge \frac { n }{ \frac { 1 }{ {x}_{1} }+\frac { 1 }{ {x}_{2} }+\cdots+\frac { 1 }{ {x}_{n} } }$$

Average Speed

If you travel a distance $S$ at speed $a$ and return at speed $b$, then the average speed $v$ is expressed as the harmonic mean of the two speeds. $$v = \frac { 2ab }{ a+b }$$

Upper and Lower Bounds of the Harmonic Mean

The harmonic mean of $a,b > 0$ is between the values of $a$ and $b$. $\max$ and $\min$ indicate the maximum and minimum values. $$\min (a,b) \le H (a,b) \le \max (a,b)$$

Explanation

Why is it called “Harmonic”?

Most Koreans first encounter the harmonic mean in high school, but quickly lose interest due to its peculiar formula and rarity even in exam studies. Above all, the naming is most difficult to accept. While the concept of the average is easy to grasp, it’s hard to imagine why the result of multiplying the sum of reciprocals by 2 $2 \left( {{ 1 } \over { a }} + {{ 1 } \over { b }} \right)$ is called the harmonic mean intuitively.

Mathematically, adding the word Harmonic to the operation of taking reciprocals like $\displaystyle {{ 1 } \over { a }}$ is common, with the harmonic series $\displaystyle \sum_{n=1}^{\infty} {{ 1 } \over { n }}$ being an example. The question then arises, ‘Why are such things called harmonic?’

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The following content feels like something read before, but it’s a hassle to look for well-organized documents now, so I’m just jotting down from memory. Don’t take it too seriously, believe it or not, I don’t feel the need to be too accurate.

… Long ago when Pythagoras couldn’t accept the existence of irrational numbers, the Pythagorean school believed that everything in the world could be explained with mathematics and that all numbers were rational. At that time, it was widely known that some strings sounded good when played together on string instruments, regardless of the instrument’s size, a phenomenon that was reproducible anywhere as long as the ratio of the lengths of those strings matched, with the ratio being fractions―which was most diligently studied by the mathematicians of the time. Even in middle school, we learn about ‘similarity of triangles’ and solve those problems by setting up ‘proportions’.

On the other hand, among the combinations of sounds produced by plucking several strings, the ‘good sounding combinations’ were called harmonies, and the Greeks called it Harmony, derived from the goddess of harmony, Harmonia in Greek mythology. If this explanation is convincing, it somewhat makes sense why the words harmonic are attached to fractional forms, reciprocals, etc.

The following complex-looking formula is called the Fourier series, and the field that further investigates things like the Fourier transform is called Harmonic Analysis. $$\begin{align*} \lim \limits_{N \rightarrow \infty} S^{f}_{N}(t) &= \lim \limits_{N \to \infty}\left[ \dfrac{a_0}{2}+\sum \limits_{n=1}^{N} \left( a_{n} \cos \dfrac{n\pi t}{L} + b_{n}\sin\dfrac{n\pi t}{L} \right) \right] \\ &= \dfrac{a_0}{2}+\sum \limits_{n=1}^{\infty} \left( a_{n} \cos \dfrac{n\pi t}{L} + b_{n}\sin\dfrac{n\pi t}{L} \right) \end{align*}$$ The fractions $\displaystyle {{ n \pi } \over { L }}$ inside the cosine and sine are commonly referred to as frequency, which can be seen as a trace of the mathematical history passed down since the Pythagorean school.

Geometric Meaning

$$\overline{XY} = {{ 2 \overline{AE} \cdot \overline{BF} } \over { \overline{AE} + \overline{BF} }}$$ There are actually a few more examples if you look into it¹, but mathematically, beyond the fact itself, I’m not really sure what more value it has. Frankly, even the graphic above, from the author’s perspective, rather explains how the harmonic mean is applied geometrically than explains the geometric meaning of the harmonic mean itself.

Proof

Upper and Lower Bounds of the Harmonic Mean

Strategy: While it’s obvious that the ‘average’ $H(a,b)$ falls between $a$ and $b$ regardless of how it’s calculated, let’s just prove the case of $\min (a,b) \le H(a,b)$ without relying on the intuition provided by the naming. Use proof by contradiction.

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If $a=b$, then its harmonic mean is trivially $2ab / (a+b) = a = b = \min (a,b)$, so let’s consider the case of $a \ne b$. Without loss of generality, assume $a < b$, then $\min \left( a,b \right) = a$, and assume $H(a,b) < \min \left( a,b \right)$. Then, $$\begin{align*} & H(a,b) < \min \left( a,b \right) \\ \implies & {{ 2ab } \over { a+b }} < a \\ \implies & 2b < a + b \\ \implies & b < a \end{align*}$$ which contradicts $a < b$, hence it must be that $H(a,b) \ge \min \left( a,b \right)$.

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