Variants and Generalizations of Cauchy

Try this

Merge the three cards one after another until only one is left. What are its numerator and denominator in terms of the original cards?
Every merge either leaves the total unchanged or makes it smaller. Which merge changed nothing? How are the ratios $x_{i} / y_{i}$ of those two cards related?
Drag the sliders so that all three ratios are equal. Does merging lose anything now?
Switch to “cube / square” and merge again. Does the pattern still hold?

Merging cards is the fraction form of the Cauchy inequality (also called the $m = 1$ case of the power-sum inequality, or Titu's lemma): for $y_{i} > 0$ ,

\frac{x _{1} ^{2}}{y _{1}} + \frac{x _{2} ^{2}}{y _{2}} + \dots + \frac{x _{n} ^{2}}{y _{n}} \geq \frac{( x _{1} + x _{2} + \dots + x _{n} ) ^{2}}{y _{1} + y _{2} + \dots + y _{n}},

with equality if and only if $\frac{x _{1}}{y _{1}} = \frac{x _{2}}{y _{2}} = \dots = \frac{x _{n}}{y _{n}}$ . It is just the Cauchy inequality for the two lists $\frac{x _{i}}{y _{i}}$ and $y_{i}$ : their products are $x_{i}$ , and their squares are $\frac{x _{i}^{2}}{y _{i}}$ and $y_{i}$ .

Predict first

The cards $\frac{1 ^{2}}{1}, \frac{2 ^{2}}{1}, \frac{3 ^{2}}{2}$ add up to $9.5$ . What is the value after merging them all into one card?

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

Why: merging two cards

Merging can only make it smaller

Step 1 / 6

Start with two cards
Look only at the first two cards, $\frac{x _{1}^{2}}{y _{1}}$ and $\frac{x _{2}^{2}}{y _{2}}$ . We want to show that merging them into $\frac{( x _{1} + x _{2} ) ^{2}}{y _{1} + y _{2}}$ does not increase the value.

$\frac{x _{1}^{2}}{y _{1}} + \frac{x _{2}^{2}}{y _{2}} \geq \frac{( x _{1} + x _{2} ) ^{2}}{y _{1} + y _{2}} ?$
Subtract directly
The loss depends on the ratios
Merge one card at a time
When nothing is lost at all
Higher exponents: the power-sum inequality

A family of relatives

Write the same idea in different ways and you get several standard variants. Each one comes from choosing the two lists well:

Fraction form (the one above): $\sum \frac{x _{i}^{2}}{y _{i}} \geq \frac{( \sum x _{i} ) ^{2}}{\sum y _{i}}$ , $y_{i} > 0$ .
When the numerator is not a square: for $a_{i}, b_{i} > 0$ , write $\frac{a _{i}}{b _{i}}$ as $\frac{a _{i}^{2}}{a _{i} b _{i}}$ to get $\sum \frac{a _{i}}{b _{i}} \geq \frac{( \sum a _{i} ) ^{2}}{\sum a _{i} b _{i}}$ , with equality if and only if all the $b_{i}$ are equal.
Hölder's inequality: for $p, q > 1$ with $\frac{1}{p} + \frac{1}{q} = 1$ , $\sum a_{i} b_{i} \leq (\sum a_{i}^{p})^{1/ p} (\sum b_{i}^{q})^{1/ q}$ ( $a_{i}, b_{i} > 0$ ). The case $p = q = 2$ is Cauchy.
The power-sum inequality: the last step above, with exponents $m + 1$ and $m$ .

Practice

Exercise 1Basic

Positive numbers $x, y, z$ satisfy $x + y + z = 2$ . Find the minimum of $\frac{1}{x} + \frac{1}{y} + \frac{4}{z}$ .

Exercise 2Basic

Real numbers $x, y, z$ satisfy $x + y + z = 6$ . Find the minimum of $x^{2} + \frac{y ^{2}}{2} + \frac{z ^{2}}{3}$ .

Exercise 3Advanced

Positive numbers $x, y$ satisfy $x + y = 1$ . Find the minimum of $\frac{1}{x ^{2}} + \frac{8}{y ^{2}}$ .

Exercise 4Challenge

Let $a, b, c > 0$ . Prove that $\frac{a ^{2}}{b + c} + \frac{b ^{2}}{c + a} + \frac{c ^{2}}{a + b} \geq \frac{a + b + c}{2}$ .

Try this

Why: merging two cards

Merging can only make it smaller

A family of relatives

Practice

Mistake Museum · Exhibits from this stop

Merging even with a negative denominator

Merging when the numerator is not a square