The Mean Inequalities · Little Blue Book, Visualized

Try this

Drag only the split point $C$ . The diameter $a + b$ does not change, so the radius (red, the arithmetic mean $A$ ) stays put. When is the blue half-chord (the geometric mean $G$ ) longest?
Set $a$ to $0.5$ and $b$ to $10$ . Which marker on the number line hugs the smaller number? Which one gets pulled toward the larger number?
However you drag, does the order of the four markers ever change? What happens to them when you make $a = b$ ?
Look at the $n$ -number version below. Make one of the numbers very large or very small: how do the gaps between the four markers change?

For positive numbers $a_{1}, a_{2}, \dots, a_{n}$ , the four means are

H_{n} = \frac{n}{\frac{1}{a _{1}} + \dots + \frac{1}{a _{n}}}, G_{n} = n a_{1} a_{2} \dots a_{n},

A_{n} = \frac{a _{1} + \dots + a _{n}}{n}, Q_{n} = \frac{a _{1}^{2} + \dots + a _{n}^{2}}{n} .

What you observed is the mean inequality (the star of this volume is the middle one, $A_{n} \geq G_{n}$ ):

H_{n} \leq G_{n} \leq A_{n} \leq Q_{n}, with equality if and only if a_{1} = a_{2} = \dots = a_{n} .

The four-number version. Make all four numbers equal and the four markers coincide.

Predict first

Let $a = 1$ and $b = 9$ . Which of these is closest to their harmonic mean $H$ ?

1.8

3

5

6.4

Choose an answer first; then you can see the result.

Why

For two numbers the algebra takes one line, and the geometry is a single semicircle picture.

The four means of two positive numbers

Step 1 / 7

Algebra: one square
For $a, b \geq 0$ a square is never negative. Expanding it gives $A \geq G$ , with equality if and only if $a = b$ , that is, $a = b$ .

$(a - b)^{2} \geq 0 ⟺ \frac{a + b}{2} \geq ab$
The radius is the arithmetic mean
The half-chord is the geometric mean
The half-chord is no longer than the radius
Project once more: the harmonic mean
A slanted segment: the quadratic mean
Equality: four segments become one

For $n$ numbers, $A_{n} \geq G_{n}$ can no longer be read off a flat picture. The next stop gives several proofs.

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: $\frac{a + b}{2} \geq ab$

Two non-negative numbers: however you drag the sliders, “The chain” shows ✓.

Practice

Exercise 1Basic

Let $a = 4$ and $b = 9$ . Find the value of $\frac{a + b}{2} - ab$ .

Exercise 2Basic

Find the harmonic mean of $3$ and $6$ .

Exercise 3Advanced

Let $a, b, c > 0$ . Prove that $(a + b) (b + c) (c + a) \geq 8 ab c$ , and say when equality holds.

Exercise 4Challenge

Four positive numbers have arithmetic mean $5$ and geometric mean also $5$ . Find the sum of their squares.

Mistake Museum · Exhibits from this stop

Before you leave, look at the traps other people fell into here.See every exhibit in the museum →

Exhibit

Forgetting the “non-negative” condition

For every

x \neq = 0

x + \frac{1}{x} \geq 2 x \cdot \frac{1}{x} = 2

What went wrong? Think first, then open

Diagnosis The mean inequality only holds for non-negative numbers. At $x = - 1$ the left side is $- 2$ , far below $2$ . The arithmetic mean of two negative numbers is negative, while the geometric mean $ab$ is positive.

Split into cases: for

x > 0

x + \frac{1}{x} \geq 2

; for

x < 0

, apply the inequality to

- x > 0

and get

x + \frac{1}{x} \leq - 2

“Before using AM–GM, make sure every term is non-negative”

Exhibit

Writing “≥” without saying when equality holds

a, b > 0

and

a + b = 4

, so

ab \leq 2

and the maximum of

ab

4

. (No word on when it is reached.)

What went wrong? Think first, then open

Diagnosis Here it happens to be right: it is reached at $a = b = 2$ . But “ $\leq$ ” only says that $4$ is an upper bound. Whether it is the maximum depends on whether equality can actually hold. The next stop and Chapter 2 show many bounds that cannot be reached.

Every time you use the mean inequality, add a sentence “equality holds if and only if …”, and check that this condition is compatible with the other conditions of the problem.

“An inequality is only half the story; the equality condition is the other half”