AM–GM with Parameters · Little Blue Book, Visualized

What this move does

The two-term mean inequality can carry a positive parameter $λ$ :

2 ab \leq λ a^{2} + \frac{1}{λ} b^{2} (λ > 0), with equality if and only if b = λa .

It holds for every $λ$ (it is just $(λ a - \frac{b}{λ})^{2} \geq 0$ ). The parameter does not change whether it is true; it only changes where equality happens. So:

Split the expression into a few parts and bound each with a parametrized mean inequality. The bound you get holds for every value of the parameter.
Add up the steps and write down the final bound (it depends on the parameter).
Solve an equation to choose the parameter so that the equality conditions of all the steps are compatible (typically: several coefficients are equal). Only then is the bound the extreme value.

Predict first

For $2 ab \leq λ a^{2} + \frac{1}{λ} b^{2}$ to reach equality when $b = 3 a$ , what should $λ$ be?

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

Work through one

Example: the maximum of (xy + 2yz)/(x² + y² + z²)

Step 1 / 6

Why split y²?
$x y$ has to be bounded by $x^{2}$ and $y^{2}$ , and $y z$ by $y^{2}$ and $z^{2}$ : $y^{2}$ is shared by both. Give it to them in the ratio $t : (1 - t)$ .

$x^{2} + y^{2} + z^{2} = (x^{2} + t y^{2}) + ((1 - t) y^{2} + z^{2})$
AM–GM on each group
Enlarge to a common coefficient
Turned too far
Make the three coefficients equal
Equality is reached

Replace the $2$ in the problem by a general $q$ , and the same computation gives $t = \frac{1}{1 + q ^{2}}$ and the maximum $\frac{1 + q ^{2}}{2}$ . Drag $q$ in the workbench at the top to check.

Practice

Exercise 1Basic

For $2 ab \leq λ a^{2} + \frac{1}{λ} b^{2}$ to reach equality when $b = 2 a$ , what is $λ$ ?

Exercise 2Advanced

Let $x, y, z$ be real numbers, not all $0$ . Let $M$ be the maximum of $\frac{x y + y z}{x ^{2} + y ^{2} + z ^{2}}$ . Find $M^{2}$ .

Exercise 3Advanced

Positive numbers $x, y$ satisfy $x + y = 1$ . Find the minimum of $x^{2} + 2 y^{2}$ .

Exercise 4Challenge

Let $x, y, z, w$ be real numbers, not all $0$ . Find the maximum of $\frac{x y + y z + z w}{x ^{2} + y ^{2} + z ^{2} + w ^{2}}$ .

Mistake Museum · Exhibits from this stop

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Exhibit

Bounding pair by pair without a parameter

x y \leq \frac{x ^{2} + y ^{2}}{2}

and

2 y z \leq y^{2} + z^{2}

; adding,

x y + 2 y z \leq \frac{1}{2} x^{2} + \frac{3}{2} y^{2} + z^{2} \leq \frac{3}{2} (x^{2} + y^{2} + z^{2})

, so the maximum is

\frac{3}{2}

What went wrong? Think first, then open

Diagnosis Every step is true, but equality cannot be reached. The first two steps need $x = y = z$ , and the last step enlarges $\frac{1}{2} x^{2}$ and $z^{2}$ to $\frac{3}{2} x^{2}$ and $\frac{3}{2} z^{2}$ , so equality there needs $x = z = 0$ , and then $y$ is $0$ as well. The true maximum is $\frac{5}{2} \approx 1.118$ .

Split

y^{2}

in an unknown ratio

t : (1 - t)

, and choose

t

so that the three square terms get equal coefficients:

t = \frac{1}{5}

“Wherever the coefficients are unbalanced, that is where the slack hides”