The AM-GM inequality says that
harmonic mean ≤ geometric mean ≤ arithmetic mean.
But it's not always clear why
it should be so. The calculus
aspects are well covered by the Jensen inequality
proof utilising convexity
), or using various versions of Lagrange multiplier
s. For 2 numbers, it's an isoperimetric inequality
that says that of the rectangles of given perimeter
has the greatest area
. But for n
numbers, it's not clear what it says.
Here's a proof of just the HM<=AM bit which makes sense physically (and can also be made completely rigourous). Any extension to cover the GM will be very much welcomed by the author! I came up with it on my own. I'm unaware of anyone else's work along these lines -- please do inform me if I'm unoriginal.
Suppose we have resistors of values R1,...,Rn. Arrange them in two ways:
- The "series parallel" network
- This has n parallel columns of resistors arranged in series, so it has effective resistance equal to the harmonic mean of the resistances:
Each of the n identical parallel block has effective resistance 1/(1/R1+...+1/Rn), so the total resistance is n times that, or the harmonic mean.
+-R1-+ +-R1-+ +-R1-+
| | | | | |
+-R2-+ +-R2-+ +-R2-+
| . | | . | | . |
--+ . +--+ . +--...--+ . +--
| . | | . | | . |
+-Rn-+ +-Rn-+ +-Rn-+
- The "parallel series" network
- This has n parallel rows of arrangements of all the resistors in series, so it has effective resistance equal to the arithmetic mean of the resistances:
Each of n identical rows has resistance R1+...+Rn, so the effective resistance is 1/n of the resistance of each row, or the arithmetic mean of the resistances.
--+ ... +--
We want to show the effective resistance of the first is less than or equal to that of the second. To do this we rearrange
each row in the "parallel series" network, so that every resistor appears once in each column:
--+ ... +--
Obviously, the effective resistance of this "revised parallel series" network is unchanged by this exercise, and it looks as though we're just pushing pieces of paper around.
BUT! "Recall" the obviously true (although not so easy to prove) theorem of electrical networks that shorting 2 points in an electrical network cannot increase its effective resistance (in fact, it is true that the effective resistance of a network of resistors is a monotone increasing function of the resistances; here we're just using a special case that lets us change resistance ∞ to resistance 0).
Armed with this theorem (and a soldering iron), let's start proving! Run a short between every two adjacent columns of the "revised parallel series" network (i.e.: Run the first wire from the point between R1 and R2 on the first row to the point between R2 and R3 on the second row to ... to the point between Rn and R1 on the last row; this takes care of the first column. Continue running wires down the remaining columns). Since we're just shorting pairs of points, the aforementioned theorem says we never increase the effective resistance.
But at the end of the process we're left with the "series parallel" network, which has effective resistance equal to the harmonic mean of R1,...,Rn. So the harmonic mean is no greater than the arithmetic mean.