Substitution in Inequalities

Normalize, Ravi, and trig substitution to reach AM-GM form.

The idea

Substitution is the standard method for bringing a hard inequality into the reach of AM-GM or Cauchy-Schwarz: rename the variables so that a constraint disappears or a familiar form appears, prove the new inequality, and translate back. Three substitutions cover most problems.

Normalization applies to a homogeneous inequality, one in which scaling every variable by $t > 0$ multiplies both sides by the same power of $t$.

Proposition (Normalization).

A homogeneous inequality in positive variables holds for all values of the variables if and only if it holds whenever the variables sum to $1$.

Its truth cannot depend on the overall scale, so fixing the scale removes a degree of freedom at no cost.

The Ravi substitution removes a triangle constraint.

Proposition (Ravi substitution).

Positive numbers $a, b, c$ are the side lengths of a triangle if and only if $a = y+z$, $b = z+x$, $c = x+y$ for some $x, y, z > 0$.

The triangle inequalities such as $a < b + c$ hold exactly when $x, y, z > 0$, so three coupled conditions become one.

Trigonometric substitution trades an algebraic constraint for an identity. A constraint such as $a^{2} + b^{2} = 1$ or $ab = 1$ becomes a relation between angles once the variables are written as sines or tangents, and the identity then carries the argument.

In every case, choose the substitution by the constraint it eliminates, not by the expression it shortens.

Ways to work on it

• Walkthrough. Three standard substitutions that turn a constrained inequality into a friendlier one.
• Practice. Compute the triangle substitution's new variables from given side lengths.
• Hardest. Prove a classic three-variable cyclic inequality by substitution.

Not sure where to start? Take the ten-question placement test.