Cauchy-Schwarz and AM-GM Inequalities

What you will get from this section. You will learn two useful forms of the Cauchy-Schwarz inequality and the two-variable AM-GM inequality. More importantly, you will learn the kinds of expressions that should make you think of using them. Neither inequality is difficult to prove in the forms we need. In fact, both ultimately come from the very familiar fact that a square cannot be negative.

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Cauchy-Schwarz inequality

For our purposes, we will discuss the following form of the Cauchy-Schwarz inequality:

$$(a^2+b^2)(x^2+y^2)\ge (ax+by)^2.$$

Proof

The proof is based on the fact that a square is always non-negative. We consider the difference of the two sides:

which is a square and is therefore greater than or equal to $0$. So

Note that equality occurs exactly when $(ay-bx)^2=0$, so when $ay=bx$, or when the pairs $(a,b)$ and $(x,y)$ are proportional.

The general Cauchy-Schwarz inequality

Cauchy-Schwarz extends to any number of real numbers. For real numbers $a_1,\ldots,a_n,x_1,\ldots,x_n$,

Equality occurs when the two lists of numbers are proportional.

We will not prove the general result here, but it is included here for completeness.

Example 1

Suppose $x^2+y^2=1$. Find the greatest possible value of $7x-24y$.

Cauchy-Schwarz gives

where the last step uses $x^2+y^2=1$. Hence

The upper bound is attained when $(x,y)$ is a positive multiple of $(7,-24)$, which is clearly possible by inspection, without needing to find the exact values of $x$ and $y$. Therefore the greatest possible value of $7x-24y$ is $25$.

Remark: This problem can alternatively be solved geometrically as finding the largest $k$ for which the line $7x-24y=k$ still meets the circle $x^2+y^2=1$. This occurs when the line is tangent to the circle.

Example 2

For positive real numbers $x$ and $y$, prove that

Apply Cauchy-Schwarz to the pairs

We obtain

This example illustrates the flexibility of the Cauchy-Schwarz inequality: it can be applied with many different choices of $a$, $b$, $x$ and $y$, which is what makes it useful in a wide range of inequality problems.

Example 3

For $x,y\ge 0$, show that

Apply Cauchy-Schwarz to the pairs $(1,1)$ and $(\sqrt{x},\sqrt{y})$:

Therefore

Both sides are non-negative, so taking square roots gives

Example 4

The point $P$ has coordinates $(1,2)$. Find the shortest distance from $P$ to the line

Let $(x,y)$ be any point on the line. Its distance from $P$ is

Since $3x+4y=20$,

By Cauchy-Schwarz,

Hence

so

Equality occurs when $(x-1,y-2)$ is proportional to $(3,4)$, so the bound can be attained.

Therefore the shortest distance from $P$ to the line is $\frac{9}{5}$.

Remark: This problem can, of course, also be solved by finding the line through $P$ perpendicular to the given line and then finding their point of intersection. Here, Cauchy-Schwarz did the heavy lifting without us having to approach the problem geometrically.

AM-GM inequality

Here is the AM-GM inequality for two positive real numbers $a$ and $b$,

$$\frac{a+b}{2}\ge \sqrt{ab}.$$

The name AM-GM simply refers to the fact that the arithmetic mean $\frac{a+b}{2}$ is at least as large as the geometric mean $\sqrt{ab}$.

Proof

Since $a,b>0$, both $\sqrt a$ and $\sqrt b$ are real. Therefore

Expanding gives

Equality occurs when $\sqrt a=\sqrt b$, so when $a=b$.

The general AM-GM inequality

AM-GM extends to any number of positive real numbers. For $a_1,a_2,\ldots,a_n>0$,

Equality occurs when $a_1=a_2=\cdots=a_n$.

Again, we will not prove the general result here, but it is included here for completeness.

Example 5

For all positive real numbers $a$ and $b$, the inequality

holds. Find the greatest possible value of the constant $c$.

Viewing $a^3+b^3$ as a polynomial in $a$, notice that $a=-b$ is a root, so $(a+b)$ is a factor, and we can factorise the left-hand side:

Since $a,b>0$, we may divide by the positive quantity $ab(a+b)$ to obtain

By AM-GM,

Therefore

Equality occurs when $\frac{a}{b}=\frac{b}{a}$, which for positive $a$ and $b$ gives $a=b$. Therefore the greatest possible value of $c$ is $1$.

Example 6

Positive real numbers $x$, $y$, and $z$ satisfy

Find the maximum possible value of $xyz$.

Using the three-variable AM-GM inequality,

Since $x+y+z=12$,

Cubing both sides gives

Equality occurs when $x=y=z$. Since their sum is $12$, equality occurs at

Therefore the maximum possible value of $xyz$ is $64$.

Cauchy-Schwarz or AM-GM?

There is some overlap between what these inequalities can do, and occasionally a problem can be solved using either one. However, as a rough guide:

• Think Cauchy-Schwarz when you see two sums of squares together with a linear combination, or when an expression can be rearranged into that structure.
• Think AM-GM when you see positive quantities whose sum or product is fixed, especially expressions such as $x+\frac{k}{x}$.

Do not try to force either inequality into every optimisation problem. Completing the square, factorisation, or ordinary quadratic reasoning may still be simpler.

Summary

The two-variable Cauchy-Schwarz inequality for all real numbers $a,b,x,y$ is

$$(a^2+b^2)(x^2+y^2)\ge (ax+by)^2.$$

The two-variable AM-GM inequality for $a, b>0$ is

$$\frac{a+b}{2}\ge \sqrt{ab}.$$

The general Cauchy-Schwarz inequality for all real numbers $a_1,\ldots,a_n,x_1,\ldots,x_n$ is

$$(a_1^2+a_2^2+\cdots+a_n^2)(x_1^2+x_2^2+\cdots+x_n^2) \ge (a_1x_1+a_2x_2+\cdots+a_nx_n)^2.$$

The general AM-GM inequality for $a_1,\ldots,a_n>0$ is

$$\frac{a_1+a_2+\cdots+a_n}{n}\ge \sqrt[n]{a_1a_2\cdots a_n}.$$

For AM-GM, equality occurs when all the numbers are equal. For Cauchy-Schwarz, equality occurs when the two lists of numbers are proportional.

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